Wheat Stone Bridge Non Linearity
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VISHAY MICRO-MEASUREMENTS Errors Due to Wheatstone Bridge Nonlinearity T E C O T E Strain Gages and Instruments Tech Note TN-507-1 For technical support, contact [email protected] www.vishaymg.com 77 Revision 02-Mar-07 1.0 Introduction Commercial static strain indicators and signal condition- ers vary considerably in their circuit details; and, although most of them are based upon some form of the Wheats tone bridge circuit, the bridge circuit is employed in differing ways in different instruments. Because of the many varia- tions in instrument design, a completely general treatment of instrument nonlinearities is not practicable within the scope of this publication. There is, however, a large class of static strain indicators and signal conditioners with a more- or-less characteristic circuit arrangement (employing the “unbalanced” Wheatstone bridge), and displaying a char- acteristic nonlinearity. This Tech Note has been prepared to provide a simple means for determining the magnitudes of the nonlinearity errors and for making corrections when necessary. Note that the error and correction relationships given here apply only to instruments having the characteris- tics defined in Section 2.0. For other instruments, the non- linearity errors, if they exist, will have to be determined by direct calibration or from manufacturers’ specifications. The nonlinearity error occurs because, when strain mea- surements are made with an “unbalanced” Wheatstone bridge circuit (as described in Section 2.0), there are certain conditions under which the output of the bridge circuit is a nonlinear function of the resistance change(s) producing that output. The error due to the nonlinearity, when present, is ordinarily small, and can usually be ignored when measur- ing elastic strains in metals. However, the percentage error increases with the magnitude of the strain being measured, and can become quite sign ificant at large strains (for example, the error is about 0.1% at 1000 , 1% at 10 000 , and 10% at 100 000 ; or, as a convenient rule of thumb, the error, in percent, is approximately equal to the strain, in percent). 2.0 The “Unbalanced” Wheatstone Bridge Circuit Most static strain indicators and signal conditioners for use with resistance strain gages use a form of the Wheatst one bridge circuit in which the bridge arms consist of one to four active gages. The classical Wheatstone bridge arrangement has been used for many years for the accurate measurement of a single unknown resistance; and, in such instruments, the bridge is balanced at the time of measurement by adjusting the resistances of the other arms. The bridge circuit found in most strain indicators, on the other hand, is unbalanced by the varying gage resistance(s) at the time of making the measurement, and is theref ore commonly referred to as the “unbalanced” Wheatstone bridge. The output voltage obtained from the “unbalanced” Wheatstone bridge is a function of the amount of unbal- ance, and is therefore directly related to the strain applied to the strain gage. However, under certain conditions frequent- ly encountered in actual practice , the bridge output voltage is, as noted earlier, a nonlinear function of the resistance change in the bridge arms. When this occurs, the strain read- ings will be somewhat in error. Figure 1 shows two of the circuit arrangements most commonly employed in commercial strain indicators and signal conditioners. In circuit (A), the bridge output voltage is amplified and displayed on an indicating instrument, fre- quently a digital voltmeter . In circuit (B), the bridge output voltage is “nulled” by an equal and opposite voltage injected into the measurement circuit. In both cases, the nonlinearity errors are identical if the amplifiers have high input imped- ances, and if the power supplies are of the constant-volt age type. Note also that in both circuits the “balance” control is R1 R2 R3 R4 E Amplifier Display Balance Control Figure 1. Circuit arrangements for commercial instruments. (A) (B)
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V I S H AY M I C R O - M E A S U R E M E N T S
Errors Due to WheatstoneBridge Nonlinearity
TE
CH N
OTE
Strain Gages and Instruments Tech Note TN-507-1
For technical support, contact
www.vishaymg.com
77
Revision 02-Mar-07
1.0 Introduction
Commercial static strain indicators and signal condition-ers vary considerably in their circuit details; and, althoughmost of them are based upon some form of the Wheatstonebridge circuit, the bridge circuit is employed in differingways in different instruments. Because of the many varia-tions in instrument design, a completely general treatment
of instrument nonlinearities is not practicable within thescope of this publication. There is, however, a large class of static strain indicators and signal conditioners with a more-or-less characteristic circuit arrangement (employing the“unbalanced” Wheatstone bridge), and displaying a char-acteristic nonlinearity. This Tech Note has been preparedto provide a simple means for determining the magnitudesof the nonlinearity errors and for making corrections whennecessary. Note that the error and correction relationshipsgiven here apply only to instruments having the characteris-tics dened in Section 2.0. For other instruments, the non-linearity errors, if they exist, will have to be determined bydirect calibration or from manufacturers’ specications.
The nonlinearity error occurs because, when strain mea-surements are made with an “unbalanced” Wheatstonebridge circuit (as described in Section 2.0), there are certainconditions under which the output of the bridge circuit isa nonlinear function of the resistance change(s) producingthat output. The error due to the nonlinearity, when present,is ordinarily small, and can usually be ignored when measur-ing elastic strains in metals. However, the percentage errorincreases with the magnitude of the strain being measured,and can become quite signicant at large strains (for example,the error is about 0.1% at 1000 , 1% at 10 000 , and 10%at 100 000 ; or, as a convenient rule of thumb, the error, inpercent, is approximately equal to the strain, in percent).
2.0 The “Unbalanced”Wheatstone Bridge Circuit
Most static strain indicators and signal conditioners foruse with resistance strain gages use a form of the Wheatstonebridge circuit in which the bridge arms consist of one to fouractive gages. The classical Wheatstone bridge arrangementhas been used for many years for the accurate measurementof a single unknown resistance; and, in such instruments, thebridge is balanced at the time of measurement by adjustingthe resistances of the other arms. The bridge circuit found
in most strain indicators, on the other hand, is unbalancedby the varying gage resistance(s) at the time of making themeasurement, and is therefore commonly referred to as the“unbalanced” Wheatstone bridge.
The output voltage obtained from the “unbalanced”Wheatstone bridge is a function of the amount of unbal-ance, and is therefore directly related to the strain applied to
the strain gage. However, under certain conditions frequent-ly encountered in actual practice, the bridge output voltageis, as noted earlier, a nonlinear function of the resistancechange in the bridge arms. When this occurs, the strain read-ings will be somewhat in error.
Figure 1 shows two of the circuit arrangements mostcommonly employed in commercial strain indicators andsignal conditioners. In circuit (A), the bridge output voltageis amplied and displayed on an indicating instrument, fre-quently a digital voltmeter. In circuit (B), the bridge outputvoltage is “nulled” by an equal and opposite voltage injectedinto the measurement circuit. In both cases, the nonlinearityerrors are identical if the ampliers have high input imped-ances, and if the power supplies are of the constant-voltagetype. Note also that in both circuits the “balance” control is
R 1 R 2
R 3R 4
E
Amplier Display
BalanceControl
Figure 1. Circuit arrangements for commercial instruments.
(A)
(B)
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Errors Due to Wheatstone Bridge Nonlinearity
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used only to establish initial bridge balance before the gagesare strained, and that the balance controls do not form partof the readout circuit. This type of “balance” circuit is nor-mally provided with a limited range so as not to cause prob-lems in resolution and setting-stability; and, therefore doesnot greatly inuence the nonlinearity errors as described in
this Tech Note. To permit a rigorous treatment of the errorswithout introducing other considerations, it is assumedthroughout the following discussion that the “balance” cir-cuit is either completely disconnected, or that the control isleft at the midpoint of its range. It is also assumed that thebridge arms are nominally resistively symmetrical about an
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axis joining the output corners of the bridge;i.e., that:
(R1/R4)nom = 1 = ( R 2/R3)nom
As a result of the circuit arrangementsdescribed above, obtaining a reading from theinstrument (whether or not the process involvesnulling a meter) has no effect on the state of resistive balance within the Wheatstone bridgecircuit. Even if the Wheatstone bridge is initiallybalanced resistively so that R 1/R4 = R2/R 3, thiswill no longer be true, in general, when one ormore of the strain gages in the bridge arms arestrained. Consequently, the Wheatstone bridgeis ordinarily operated in a resistively unbalanced
state. In this mode of operation, resistancechanges in the bridge arms may cause changesin the currents through the arms, dependingupon the signs and magnitudes of the resistancechanges in all four arms. When current changesoccur, the voltage output of the bridge is notproportional to the resistance changes, andthus the output is nonlinear with strain, and theinstrument indication is in error.
3.0 Nonlinearities and Corrections
For the class of instruments described inSection 2.0, Table 1 illustrates the nonlinearities
to be expected, and includes correction relation-ships. The table gives the nondimensional out-put voltage ( E o /E ) as a function of the applied strain (andgage factor) for a variety of commonly encountered strainstates and different arrangements of gages on the structuralmember and within the Wheatstone bridge. It can readily beseen that the output expressions for cases 1, 2, 4, and 5 areintrinsically nonlinear, while those for cases 3, 6, and 7 arelinear. Examination of the column of bridge/strain arrange-ments demonstrates that only when the resistance changesdue to strain are such that the currents through the bridgearms remain constant — that is, R1/R 1 + R4 /R 4 = 0 and
R 2/R2 + R3/R3 = 0 — is the output a linear function of the strain.
For each of the cases in Table 1, the nondimensional cir-cuit output can also be expressed in the following form:
E o /E = K x 10 –3 (1– ), mV/V
In this relationship, K is a constant, determined by the gagefactor of the strain gage(s) and the number of active armsin the bridge circuit; and (when not zero) represents thenonlinearity caused by changes in the currents through thebridge arms. One hundred times the fraction /1 is then thepercentage nonlinearity in the circuit output.
The nonlinearity column in Table 1 gives the mathemati-cal expression for calculating as a function of the appliedstrain and other relevant parameters. It can be noticedfrom the table that on a percentage basis the nonlinearitymagnitudes are identical for cases 1 and 4, and for 2 and5, although the circuit outputs differ. For convenience inquickly judging nonlinearity magnitudes, the relationshipsin cases 1, 2, 4, and 5 are plotted in Figure 2, assuming posi-tive (tensile) strains, a gage factor of 2.0, and Poisson’s ratio(where involved) of 0.30. The nonlinearity for compressivestrain is opposite in sign and somewhat different in mag-nitude, but can always be calculated from the relationshipsgiven in Table 1.
The last (right most) column in Table 1 provides the rela-tionships for converting the indicated strain, i , as registeredby a strain indicator or other instrument system to the actualsurface strain under a single active gage, . The expressions inthis column correct for Wheatstone bridge nonlinearity (whenpresent) and for the number of active gages in the circuit.
4.0 Numerical Examples
For simplicity in presentation, the following three numer-ical examples assume, in each case, a quarter-bridge circuit
1.0
2.0
3.0
4.0
5.0
6.07.08.0
9.010.0
0.90.80.70.6
0.5
0.4
0.3
0.2
0.11000 10 000 100 000
Case #1, #4
Case #2, #5
P e r c e n
t N o n
l i n e a r i t y
Strain,
Figure 2. Nonlinearity errors for tensile strain in bridge circuits.
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Subsequent strain measurements at or near the calibratedstrain level do not require correction for nonlinearity.However, measurement at a signicantly different level willbe somewhat in error due to the different nonlinearity at adifferent strain level. Tech Note TN-514 also gives relation-ships for adjusting indicated strains to account for calibra-tion at one level, followed by strain measurement at a dif-ferent level. Although not treated here, leadwire resistanceis another factor to be considered in shunt calibration, andTech Note TN-514 provides relationships to correct forleadwire effects.
Errors can also arise when the initial state of Wheatstonebridge unbalance is different during shunt calibration than itis when strain measurements are to be made. If this situationexists, it is necessary to measure the initial unbalance and
determine the actual simulated strain following the proce-dures demonstrated in the second and third of the precedingnumerical examples.
Whenever dynamic strain measurements are made with aWheatstone bridge circuit, the bridge is always operated inthe unbalanced mode. Therefore, the nonlinearities listed inTable 1 of this Tech Note apply to every such dynamic strain
measurement — assuming, again, that the bridge is initiallybalanced resistively. Under these conditions, the error due tothe nonlinearity is ordinarily small at typical working strainlevels, as illustrated by Figure 2. However, if the bridge isinitially unbalanced, the nonlinearity error can be muchgreater; and, with large initial unbalances, may result in sig-nicantly inaccurate strain indications.
6.0 Summary
The nonlinearity errors occurring in conventional straingage bridge circuits are normally small enough to ignorewhen measuring modest strain magnitudes such as thoseencountered in the elastic range of metals (if the bridge isinitially balanced resistively). Large resistive unbalancescan, on the other hand, lead to sizable errors in strain indi-
cation. The relationships and procedures presented in thisTech Note can be used when necessary to correct for sucherrors. It also follows that for accurate strain measurements,it is imperative to select strain gages with tightly controlledresistance tolerances, and to minimize resistance shiftsduring gage bonding by carefully following recommendedinstallation techniques.
