Description
This is an informational application note provided by Pacific Instruments, Inc. to teach the basics of strain gage instrumentation. Learn more about in depth applications, common techniques and mathematical concepts involved in strain gage measurments taken with electronic instrumentation.
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Introduction
Strain gages are commonly used to investigate strain in
structures and materials. Excited by an electric current, the
resistance change of the gage due to tension or compression
produces a proportional voltage change which can be measured
with electronic instrumentation.
The Wheatstone bridge is the most convenient and common
circuit for measuring strain. Shown in Figure 1, the bridge is
configured such that one arm. the strain gage, changes resis
tance due to the application of strain. If the bridge is initially
balanced, the output is an indication of change. R1, R2 and R3
are fixed completion resistors usually equal to the nominal
strain gage resistance.
Temperature effects in strain gages may be eliminated using
the circuit shown in Figure 2. This circuit, which is the most
commonly used in strain gage work, employs two gages as
adjacent arms of the bridge. The active gage is mounted on the
stressed material and the dummy gage is mounted on an un
stressed area of the same material. In this bridge configuration,
equal changes in the active and dummy gages, due to temper
ature, cancel and the output is unaffected. Changes in the active
gage due to strain which are not present in the dummy gage will
produce an output. The effects of temperature have practically
no net effect on the output, except to the extent that the dummy
gage measures thermally induced stresses in the material.
It can be beneficial to make both gages active. For example, two
gages can be used on opposite sides in measuring the bending
of a beam. Applied force will cause a tension strain in one gage
and compression strain in the other. Connected as shown in
Figure 3, the bridge will be unbalanced in proportion to the
difference in the strains acting on the two gages. Additionally,
assuming that the two gages are subject to the same temperature,
temperature effects are cancelled.
All four arms of the Wheatstone bridge may be made up of
strain gages, any or all of which may be active. It is possible to
realize temperature compensation and higher output using
multiple gage configurations. When using multiple gages, one
must remember lhat the bridge is unbalanced in proportion to
the difference of the strains in gages located in adjacent arms
and in proportion to the sum of the strains of gages located in
opposite arms. Output voltage may be further increased by
operating two or more bridges with their outputs in series.
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Strain Gage Wiring
Lead wires can have a significant effect on the accuracy of
strain gage sensors. Figure 4 shows a simple two wire connection
to a self-compensated strain gage. Lead wires "a" and "b"
connect the gage to the measuring circuit in which R1, R2, and
R3 are the bridge completion resistors and equal to the gage
resistance. If lead wires "a" and "b" are of low resistance, or
made from a low temperature coefficient wire, this circuit can
produce usable results. In most cases, however, this circuit
will produce significant errors and may be unusable with copper
wire in lengths greater than 20 feet.
To demonstrate the magnitude of the error which may result
using the circuit of Figure 4, consider the case where lead wires
"a" and "b" are 22 gage copper wire, 20 feet long, and subject to
a temperature range of 20° to 50° C. The total resistance a + b
for this size wire is 0.648 ohms. The approximate temperature
coefficient of copper is 0.004/° C. The change of lead resistance
will therefore be 30 X .004 X .648 = 0.078 ohms. Assuming a 350
ohm gage, the resistance change due to effects of temperature
on the leads is 222 ppm. At a gage factor of 2, this corresponds
to a strain error of 111 microinches/inch, a considerable error in
most applications. Some method of compensating for lead wire
effects is necessary if accuracy is to be preserved.
A simple solution to this problem is shown in Figure 5. This is
the "3-wire" technique of resistance measurement. As in Figure
4, leads "a" and "b" are equal and subject to the same temperature
throughout their length. In Figure 5, the resistances of
leads "a" and "b" are in series with adjacent arms of the bridge.
Lead "c" is outside the bridge and in series with the excitation
source. The result is that any voltages in "a" and "b" due to
temperature changes cancel, producing no net effect in the
output of the bridge.
While the circuit of Figure 5 will be adequate for most practical
strain gage measurements, it should be pointed out that there is
a still more precise method of lead wire compensation employing
the Kelvin bridge or "4-wire" configuration. This method of
resistance measurement will be required in only the most critical
work, and is described in textbooks on electrical measurements.
Is could be required in situations involving very long input leads
of small diameter wire or extreme temperature fluctuations, or
where it is impossible to contain input wires in the same
temperature environment.
Figure 6 shows the 3-wire connection of a two arm installation
using either 2 active or an active and compensating gage. Since
wires "a" and "c" are of the same resistance and subject to the
same temperature, and in adjacent arms of the bridge, their net
effect will be cancelled and not appear in the output. Lead wire
"b" is in series with the excitation source and will have no effect
on the bridge output.
Four active-arm bridges use conventional wiring techniques.
When remote sensing is employed, two additional wires are
required to sense the excitation voltage at the transducer. A four
active arm bridge with remote sensing will require a minimum of
six wires without provisions for calibration. When the calibration
connections are added, 8 or 10 input wires may be required.
Since this paper deals primarily with stress analysis, we will
not cover the 4 active arm bridge in detail.
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Bridge Output (Equal Resistance)
Measurements using the Wheatstone bridge require that the
deviation of one or more resistors from an initial value be determined.
Figure 7 shows a single active-arm bridge where the
completion resistors are nominally equal to the gage resistance.
The output voltage fora fractional resistance change, X, is
indicated by the following equations. Note that white the relationship
between the bridge output and X is not linear, for small
changes of X it is sufficiently linear to ignore the error in
most applications.
Previously, we determined that the output of a bridge can be
doubled if the opposite arms change resistance in the same
direction. Opposite arms edd while adjacent arms subtract.
Following is a calculation of the bridge output for changes in
the opposite arms.
Using the equation developed for the single active arm bridge,
we can now relate bridge output to strain. The ratio of resistance
change to strain change, ohms-per-ohm divided by inches per
inch, is known as the gage factor. Expressed mathematically, the
gage factor K equals ∆R/R divided by ∆S. The fractional resistance
change X can be replaced by it's equivalent ∆R/R in the
bridge output equation. The result is an expression for output
voltage in terms of strain and the gage factor K.
Rearranging the above equation, we can derive an expression for
strain as a function of output voltage.
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Bridge Output (Unequal Resistance)
Preceeding calculations assume that all bridge arms are equal.
The following derivation does not restrict the application to
equal resistance bridge arms. Figure 8 considers the active half
of the bridge separately. The output voltage is calculated as...
To find the change in output voltage for a given change in gage
resistance Rg, the above equation is differentiated with
respect to Rg.
Multiplying and dividing the right hand member by Rg gives...
The definition of gage factor is...
Hence...
In the wheatstone bridge circuit with only one gage subject to
strain, the bridge is initially balanced such that the output
voltage is zero. As strain is applied to the active gage, the bridge
is unbalanced producing an output voltage e0. In Figure 9,
assuming balance is performed at zero strain, output is equal to
that calculated for the half bridge...
Where ∆S is the strain in gage A. and gage B is a dummy. If the
functions of gages A and B are interchanged, the same equation
applies but output is of opposite polarity. If both gages are
active, and if the sign of the strain in gage B is the same as that
in gage A. then the resulting net change in bridge output
voltage will be the difference of the voltages given by each
of the respective equations. If the signs are opposite, the net
voltage will be the sum of the two individual outputs.
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Shunt Calibration
Shunt calibration is the technique commonly used for calibrating
system gain with Wheatstone bridge circuits. One or more legs
of Ihe bridge is shunted with a known resistance producing a
specific change in bridge balance simulating a compressive
strain. The output will respond exactly as if a strain had actually
occured for the existing bridge excitation and system gain.
Sensitivity can be determined from the simulated strain and
system output.
It is important to note that shunt calibration, while verifying the
operation and calibration of the measuring circuit, in no way
verifies or calibrates the strain characteristics of the gage itself.
It is important to note that shunt calibration, while verifying the
operation and calibration of the measuring circuit, in no way
verifies or calibrates the strain characteristics of the gage itself.
Two separate leadwire effects reduce the accuracy of shunt
calibration unless proper corrections are made. Leadwire resistance
in series only with the calibration resistor is one effect and
can be easily corrected by subtracting from the calculated RCAL
value. The second effect is due to uncompensated leadwire
resistance in series with the gage or gages. Here a correction,
in the form of an artificial gage factor Ką.can be calculated in
which the actual gage factor K is corrected for leadwire
resistance.
Where:
K = gage factor
Rg = resistance of gage
Ra = resistance of uncompensated lead wire
Using the definition of gage factor, where S is the simulated
strain, we see...
Where:
∆S = simulated strain
Rg = resistance of leg shunted
Ką = effective gage factor
Replacing Rg by the parallel combination of Rg and the calibration resistor, RCAL.
Rearranging terms and converting to microstrain, we obtain...
Rearranging terms, we can develop a formula for calculating
the calibration resistor required to give a specific strain.
Using the formula developed previously for the equal resistance
bridge, we can calculate the output voltage for simulated strain.
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Practical Circuits
We can now apply the shunt calibration technique to the
"three-wire" technique of leadwire compensation. Four of the
more common circuits will be discussed. While other, more
complicated circuits exist, they are beyond the scope of this
presentation.
Figure 10a is a single active arm bridge with either a compensation
gage or completion resistor for the dummy arm. The active
arm is shunted by RCAL. While this circuit requires two additional
wires for the calibration resistor and simulates only compressive
strain, it is classcial and has no required gage factor correction.
For accurate calibration, the lead wire resistance in series
with the calibration resistance should be subtracted from the
calculated value of RCAL.
The same principal is used in Figure 10b, where the completion
resistor (dummy arm) is shunted. Leadwire correction is not
required and accuracy is independent of the gage resistance. Use
is limited in that only tension can be simulated by shunt resistance.
Some users alter this circuit to perform series insertion
thus simulating compressive strain.
Both compressive and tension strain are simulated using the
circuit in Figure 10c which is a combination of the two preceeding
circuits with no additional leadwires. While gage factor correction
is not required for tension, it must be used for compression
making application somewhat complicated. The circuits
which follow provide bipolar calibration without this
complication.
Figure 10d is a circuit for bipolar calibration of 2 and 4 active
arm bridges by shunting the active arms. It is usable with any
gage wiring technique, but requires three additional leadwires:
The resistance of gages "A" and "B" must be equal to obtain
equal + and - calibration steps. No gage factor correction is
necessary, but as in 10b resistance in series with RCAL will
alter the calculated value.
A bipolar shunt technique, which requires no additional leadwires,
shunts the bridge completion resistors as shown in Figure 10e.
A correction factor is required for K to use the output equations.
Calibration is independent of the gage resistance. Recommended
for 1 and 2 active arm gages. Has reduced accuracy when used
with 4 active arm bridges.
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Contact
Pacific transducer signal conditioning amplifers provide unparalleled performance and accuracy for strain gage instrumentation.
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For additional information about strain gage instrumentation, please feel free to contact us at:
Phone: 925-827-9010
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