Electrical esistance Strain Gae Circuits In the precedin sections we have examined the various construction schemes and application procedures for electrical resistance strain aes. By these techniques and devices, then, we have supposedly succeeded in transferrin the extensional strain of the surface of a specimen into a proportional (or equal) strain in the element(s) of a resistive strain ae. The problem et this juncture is to develop a method suitable for the conversion of resistance chanes in the ae element into a more readily observable quantity. We must, in other words, transduce resistance chanes. At first consideration, resistance measure appears to be a straihtforward operation. It is a basic electrical parameter and ohmmeters for its measure are a common electronic instruments. In the present application, however, the accuracy and resolution needed for the measure of strain at enineerin levels is beyond the capabilities of all but the most refined of these instruments. As a result, special circuitry and wirin procedures have been developed to directly and accurately convert the resistance chanes into proportional voltae chanes. Several techniques will be discussed in the followin sections. Electrical Properties of the esistance Gae The fundamental formula for the resistance of a wire with uniform cross section, A, and resistivity, ρ, can be expressed as: L = ρ () A where L is the wire lenth. This relation is enerally accurate for common metals and many L L nonmetals at room temperature when subjected to direct or low frequency currents *. We consider the ae to be formed from a lenth of uniform wire and subjected to an elonation as shown: The chane in resistance can be expressed from Eq. as L L + L = ρ ( ρ + ρ) A A + A where sinifies a chane in the quantity. This is a complicated expression in its present form, however, it should be clear that for metallic wires subjected to enineerin strain levels that L << L and A << A. If ρ<< ρ as well, then we can simplify the expression by approximatin with the infinitesimal differential chane, d(): * esistivity can also be affected by electromanetic, nuclear, barometric optical and surface effects. AE5 esistance Strain Gae Circuits Pae
L d = d ρ A The differential expression on the riht side is tedious to compute directly but can be easily determined usin lo derivatives as follows. First take the natural lo (ln) of the equation yieldin: ln( ) = ln( ρ) + ln( L) ln( A) and now take the differential of this recallin that d(ln(x))=dx/x to et the simpler result: d dρ = + ρ dl L In eneral we may write A=C D where D is a cross section dimension and C is some constant (e.., D= and C=π for a circle). Usin the lo derivative method, it follows that: da dd = A D At this point we note that the lonitudinal strain can be written in differential form as: and the transverse or lateral strain as: dl ε = L ε = dd D D da A Also for linearly elastic and isotropic behavior of the wire: Then usin these results: ε D = υ ε da dl = ε D = υ ε = υ A L Finally, the resistance chane per unit resistance ( /) can then be written: d ρ = d + ( + υ ) ε () ρ This expresses the basic proportionality between resistance and strain in the ae element material. A measure of the sensitivity of the material (or its resistance chane per unit applied strain) is defined as the Gae Factor: d / GAGE FACTO = = () ε From the above resistance calculations (Eq. ) the Gae Factor can then be determined as AE5 esistance Strain Gae Circuits Pae
dρ / ρ = + υ + ε The Gae Factor as expressed above includes effects from two sources. The first term on the riht represents directly the Poisson effect, i.e., the tendency in an elastic material to contract laterally in response to axial stretchin. The second term represents the contribution due to chanes in resistivity of the material in response to applied strain. In the absence of a direct resistivity chane, then, the maximum and minimum values expected for the Gae Factor would be correspondin to the theoretically allowable rane 0 ν ½ for Poisson's atio. In practice the Gae Factor is not nearly so well behaved. This is amply evident in Fiure which is a plot of fractional resistance strain aainst strain for common ae materials. In this fiure the Gae Factor is simply the slope of the curve. The 0% hodium/platinum alloy exhibits a desirably hih but this chanes abruptly at about 0.% strain - an undesirable behavior in all but the most special cases. Pure nickel is also poorly behaved and exhibits a neative for small strain! This material is seldom used alone but is often employed as an element in other alloys. The most common material for static strain measure at room temperatures is the relatively well behaved constantan alloy. Table presents a summary of the Gae Factor and ultimate elonation for several materials. 0% rhodium/ platinum % / Ferry alloy Constantan alloy 0% old/palladium 0 Nickel % Strain % Strain Fiure. Chane in esistance with Strain for Various Strain Gae Element Materials At this point a basic difficulty has appeared. The Gae Factor is only of order unity and therefore the resistance chanes in the ae must be of the same order as the strain chanes! In enineerin materials this strain level is typically from to 0,000 microstrain or 0.00000 to 0.0. Thus, chanes in resistance in the ae of no more than l% must be detected! Herein lies the challene in the desin of measurin circuitry and a problem in practical installations. (It miht be noted with satisfaction, however, that our oriinal assumption of small / has been verified and thus the development is sound.) Two examples illustrate the point: AE5 esistance Strain Gae Circuits Pae
Table. Gae Factors for Various Grid Materials Material Gae Factor () Ultimate Elonation Low Strain Hih Strain (%) Copper.6. 0.5 Constantan*..9.0 Nickel -.7 -- Platinum 6.. 0. Silver.9. 0.8 0% old/palladium 0.9.9 0.8 Semiconductor** ~00 ~600 -- * similar to Ferry and Advance and Copel alloys. ** semiconductor ae factors depend hihly on the level and kind of dopin used. Example Assume a ae with =.0 and resistance 0 Ohms. It is subjected to a strain of 5 microstrain (equivalent to about 50 psi in aluminum). Then = ε = (5e 6)(0) = 0.00Ohms = 0.00% chane! Example Now assume the same ae is subjected to 5000 microstrain or about 50,000 psi in aluminum: = ε = (5000e 6)(0) =. Ohms = % chane esistance Measurin Circuits It is apparent thus far that quite small resistance chanes must be measured if resistance strain aes are to be used. Direct measurement of 0.00 Ohm out of 0 Ohms would require a meter with a resolution of better than one part in 00,000, in other words, a 6 diit ohmmeter! Several measurin techniques are available for this purpose but the Wheatstone Bride circuit has proven the most useful for a number of important reasons. This circuit will be described shortly but first several fundamental techniques will be discussed. Current Injection This is a simple and common technique used to make measurements of resistance accurate to about 0.% at best. A constant known current is forced throuh the unknown resistance, and accordin to Ohm s Law the resultin voltae drop across it is directly proportional to the resistance. AE5 esistance Strain Gae Circuits Pae
Constant Current Source e This technique is enerally used in portable ohmmeters and combination volt-ohm-ammeters. A major drawback for the present application is that this circuit indicates the total resistance and not the chane in resistance, which what is really desired. Ballast Circuit This circuit shown below is similar to the current injection technique but avoids the need for a constant current source. Instead, a simple voltae source is used and the ae is placed in series with a ballast resistor. This makes the voltae source resemble a constant current source (note that if B WKHXWSXWLVDOZD\VHTXDOWHLQGHSHQGHQWO\IWKHYDOXHI5 ). ballast E b e The voltae output, e, is iven as: e = + When used with strain aes, only very small chanes in resistance,, are developed and it is appropriate to make the assumption that G5 so that the chane in output voltae, de, can be determined in terms of differentials as: d de = + b b E d = = ( + ) b E ( + ) b b b ( + ) ) ε where use has been made of previous results for d / (Eq. ). The only remainin question is to determine an appropriate value for the ballast resistance, b, and this can be done by askin for a value that will maximize the sensitivity of the circuit, or in other words, provide the reatest output per unit chane in strain. The sensitivity of this circuit can be defined as: d b E () AE5 esistance Strain Gae Circuits Pae 5
de S = = ε E b ( + b ) and the maximum sensitivity is attained when b is adjusted so that: ds b 0 = = E d + This occurs when: Then b ( ) b b = de = ε E and the total output voltae is the sum of the initial (e) and the chane (de) as: e+de=e/ + / ε E As a point of interest it should be noted that these results are valid only for infinitesimally small chanes in resistance, d, induced by the applied strain, ε. If we were to consider finite values, the ballast circuit output, e+ e, is not linear with as indicated by Eq. above. ather, the relationship is nonlinear as shown in Fiure below. Note that the sensitivity calculations above indicate that the middle curve in Fiure (with b = ) provides the reatest sensitivity (larest slope) and also results in e/e=0.5 when =0. But of course, the above results are only for infinitesimal d and so represent the behavior around the zero point on the / scale in Fiure (e.., linearized behavior). 0.8 0.6 / b = e/e 0. 0. / b = / b =0. 0-0 / Fiure. Ballast Circuit NonlinearBehavior for Finite Chanes in Gae esistance, AE5 esistance Strain Gae Circuits Pae 6
Aain, as with the previous circuit, the major drawback is that the strain produces a relatively small chane in the output which is nominally E/. As an example, consider a typical confiuration with =.0, =0 Ohms and ε=5 microstrain de = (5e 6) E volts : e + de = ( + 0.000005) E volts It should be clear that such a circuit would require voltae measurement with a resolution of at least 5 parts per million so a suitable diital voltmeter would require 6 decades of precision! These difficulties can be rectified if we can devise a technique for eliminatin the steady state, E/, output. This is effectively accomplished in the well-known Wheatstone Bride which is next discussed. Wheatstone Bride The Wheatstone Bride is the most basic of a number of useful electrical bride circuits that may be used to measure resistance, capacitance or inductance. It also finds applications in a number of circuits desined to indicate resistance chanes in transducers such as resistance thermometers and moisture aes. In the circuit shown below it is apparent that the bride can be imained as two ballast circuits (composed of, and, ) connected so that the initial steady state voltaes are cancelled in the measurement of e. e e DC DC The output voltae can be written as the difference between two ballast circuits as: e = E = E (5) + + ( + )( + ) Clearly, an initial steady state voltae exists unless the numerator above is zero. Such a confiuration with zero output voltae is termed a Balanced Bride and is provided when: = This relationship is not of direct concern here but it is interestin to note that if any three of the four resistances are known, the fourth can be determined by ratioin the values obtained at balance. For the present, however, we are concerned with the output produced by small chanes in the resistance of the bride arms. If we consider infinitesimal chanes in each resistor then i i + d i and we can compute the differential chane in the output voltae, e, as: AE5 esistance Strain Gae Circuits Pae 7
or e de = de = ( + ) e d + d d d e + d + ( + ) If the bride is balanced so that: = = = = then usin this in the above equation for de yields: e + d d d d d d d de = + E (6) We can use Eq. to express de in terms of the strains as: de = [ ε ε + ε ε ]E where ε i, is the strain in the ae placed in the i-th arm of the bride. Since the bride is initially balanced, this is the only output and we have the desired result that: e = [ ε ε + ε ε ]E (7) This is the basic equation relatin the Wheatstone Bride output voltae to strain in aes placed in each arm. Several remarks are in order: The equation identifies the first order (differential) effects only, and so this is the linearized form. It is valid only for small (infinitesimal) resistance chanes. Lare resistance chanes produce nonlinear effects and these are shown in Fiure where finite chanes in ( ) in a sinle arm are considered for an initially balanced bride. Output is directly proportional to the excitation voltae and to the Gae Factor. Increasin either will improve measurement sensitivity. Equal strain in aes in adjacent arms in the circuit produce no output. Equal strain in all aes produces no output either. Fixed resistors rather than strain aes may be used as bride arms. In this case the strain contribution is zero and the element is referred to as a dummy element or ae. Equations 6 and 7 alon with the above remarks thus serve to describe the electrical behavior of a Wheatstone Bride. The major intent here is not to suest electrical measurement techniques but rather to describe the behavior in enouh detail so that the effect of chanes in any parameter can be directly assessed. While there are cases when the entire bride circuit must be custom assembled, as for example in a special transducer, a majority of typical strain ae applications in structural testin involve measurement of strain in sinle aes, one at a time, with commercial equipment. Since all presently available strain measurin instruments employ the Wheatstone Bride circuitry, our discussion on the relative effects of Gae Factor and strain chanes in individual arms on the output indication is directly applicable. E AE5 esistance Strain Gae Circuits Pae 8
0. 0 e/e -0. -0. - -0.5 0 0.5 / Fiure. Wheatstone Bride Output for Initially Balanced Bride with Sinle Active Arm Wheatstone Bride Circuit Considerations The equation describin the basic electrical response of the Wheatstone Bride circuit has been developed in the precedin section. In the followin discussions we will examine the effects of chanes in various parameters and will consider several wirin confiurations that find application in strain measurement. Temperature Effects in the Cae Fluctuations in ambient and in operatin temperatures produce the most severe effects enerally dealt with in strain measurin circuitry. The problems arise primarily from two mechanisms: () chanes in the ae resistivity with temperature and () temperature induced strain in the ae element. Additionally, for certain bride circuits in which the elements are widely separated (~ 0-00 feet), the thermally induced resistance chanes in the lead wires may also be sinificant. Typical values for the fractional chane in resistivity per deree for common strain ae element materials is presented in Table. Fiures are expressed as temperature coefficients in parts per million (ppm) rather than fractions or percentaes to better illustrate the manitude of the effect. For example, the value, 60, for isoelastic material is equivalent (usin the definition of from Eq. and =.5) to an apparent strain: ε = d / 60 0 = = 7 microstrain / F.5 It is obvious from this that chanes in ae temperature of only a few derees will produce apparent indications of hundreds of microstrain. Several alloys listed in Table have obviously AE5 esistance Strain Gae Circuits Pae 9
been chosen for their very low temperature coefficient of resistivity. The constantan alloy is probably the most common material for eneral static applications Isoelastic alloy is frequently used for aes which are subjected to dynamic strains but when interest is in the measurement of peak-to-peak values only. In this case the hiher ae factor is attractive while the thermally induced chane in resistance appears as a steady state offset and is not recorded in peak-to-peak (e.. AC) measurements. Table. Properties of Various Strain Gae Grid Materials Material Composition Use esistivity (Ohm/mil-ft) Temp. Coef. of esistance (ppm/f) Temp. Coef. of Expansion (ppm/f) Max Operatin Temp. (F) Constantan 5% NI, 55% Strain Cu Gae.0 90 6 8 900 Isoelastic 6% Ni, 8% Strain Cr, 0.5% Mo, 55.5% Fe ae (dynamic).5 680 60 800 Mananin 8% Cu, % Strain Mn, % Ni ae (shock) 0.5 60 6 Nichrome 80% Ni, 0% Thermometer Cu.0 60 0 5 000 Iridium- 95% Pt, 5% Ir Thermometer Platinum 5. 5 700 5 000 Monel 67% Ni, % Cu.9 0 00 Nickel - 5 00 8 Karma 7% Ni, 0% Strain Cr, % Al, % Fe Gae (hi temp). 800 0 500 Erroneous indications of strain can also occur in ae installations where there is a difference in the coefficients of thermal expansion of the ae and the substrate material. Precise calculation of this effect is difficult due to the variety of mechanisms which come into play. For example, if the cross-sectional area of the ae element transverse to its measurin axis is very small compared to the similar cross-section of the substrate material, then essentially all of the substrate thermal strain is transferred no the ae (a coupled-stress problem). This is enerally the case in practice and from Table it appears that for a constantan ae on an aluminum substrate with a thermal expansion coefficient of ppm/ F the contribution from this effect is: ε = 8 = 5 microstrain / Thus the net apparent strain due to resistive as well as expansion effects would be rouhly: 0 ε = + 5 = 8 microstrain / F In the usual practice the strain ae manufacturer cold works or otherwise metallurically treats the ae element material so as to equalize the thermal effects over a reasonable temperature span when the ae is mounted on a specified substrate. This is marked on the packae and a plot of error as a function of temperature, or equivalently, a plot of ae factor correction as a function of temperature is provided. Gaes typically are available compensated for use on materials with thermal expansion coefficients shown in Table. 0 F AE5 esistance Strain Gae Circuits Pae 0
Table. Materials for which Strain Gaes can be Compensated (typical) PPM/ F Material Molybdenum 6 Steel; titanium 9 Stainless steel, Copper Aluminum 5 Manesium 0 Plastics It should be pointed out here that temperature induced resistance chanes in the lead wires, especially if they are over 0 feet lon, can also produce substantial error. Gae modification is of no use but compensation can be made in the Wheatstone Bride circuitry as will be illustrated later. Temperature chanes in the aes will obviously be produced by chanes in ambient temperature. In addition sinificant temperature chanes may be produced directly in the ae by resistive heatin due to the electrical current flow. The manitude of this effect is a direct function of the power applied to the ae and the ability of the ae to dissipate the resultant heat. Gaes mounted on thin materials with poor thermal conductivity (such as plastics or fiberlass or thin sheet metal) are most affected by this problem, while conversely, aes attached to thick (compared to ae dimensions) material of hih thermal conductivity (such as copper or aluminum) are least susceptible. Experience has proven that power densities of from to 0 watts per square inch over the ae element area can be tolerated dependin on the substrate material and accuracy required. Particular recommendations are shown in Table. Table. Strain Gae Power Densities for Specified Accuracies on Different Substrates Accuracy Substrate Conductivity Substrate Thickness Power Density (r) (Watts/sq. in.) Hih Good thick -5 Hih Good thin - Hih Poor thick O.5- Hih Poor thin 0.05-0. Averae Good thick 5-0 Averae Good thin -5 Averae Poor thick - Averae Poor thin 0.-O.5 Table Notes: Good = aluminum, copper Poor = steels, plastics, fiberlass Thick = thickness reater than ae element lenth Thin = thickness less than ae element lenth The above dissipation fiures must be translated into allowable excitation levels (E) for the bride circuit accordin to the usual electrical procedures. Power enerated within a ae in a Wheatstone Bride is iven as: P = E i (8) where E and i are voltae across and current throuh a iven ae (arm). Due to the symmetry AE5 esistance Strain Gae Circuits Pae
of the Wheatstone Bride it is easy to show that one half of the total current flows throuh any arm and that one half of the power supply voltae develops across it. Thus with E and i defined as the Wheatstone Bride supply voltae and total bride current: E i P = = E i (9) but since the net bride resistance as seen by the power supply is (e.., x in one half in parallel with x in the other half) it follows (usin Eq. 8 for the complete bride) that: Finally, usin Eq. 0 in Eq. 9:: Power density is iven by E E i = = (0) P bride E = P = A P ρ = () L W where L =ae rid lenth, W =ae rid width, and P =ae power dissipation. Thus the maximum excitation voltae is iven by: E ρ L = max W () In this case, ρ must be selected from the previous table and L, W, and are determined by the particular choice of ae. One should note that hiher electric output per unit strain (e.., sensitivity) can be achieved by usin hiher excitation levels, subject to the above power limitations. Thus bride output is increased, then, by increasin: Strain (ε) Gae Factor () Gae resistance ( ) Gae element area (L W ) Temperature Compensation in the Bride Circuit Temperature compensation of the strain ae alone does not enerally eliminate thermal problems entirely. Such compensation is rarely exact and the differences must usually be eliminated by careful confiuration of the Wheatstone Bride circuit. The ability to make such compensation is, in fact, one of the more desirable features of this circuit. This is accomplished as follows. The extraneous effects of temperature and other factors inducin a resistance chane in the T ae can effectively be considered as an additional strain, ε i so that the strain in the i-th arm becomes T ε i + ε i Then the bride equation can be written: AE5 esistance Strain Gae Circuits Pae
e = T T T T [ ε ε + ε ε ] E + [ ε ε + ε ε ]E where the rihtmost term represents the extraneous temperature effects. It should be clear from this result that if the extraneous temperature effects, ε i T, are all equal, they will cancel out; in fact they will cancel out so lon as equal effects occur in any pair of adjacent arms (e.., arms and or arms and, etc). On the other hand, it should be noted that if these same effects occurred in two opposite arms only (e.., arm and arm ), the effect would be additive and would not cancel out (not desirable!). When similar strain aes are used in all four arms of the bride and when they are mounted so that each experiences the same temperature chane, then the bride output voltae will be a function of the material strain only. In this case the temperature induced resistance chanes will be the same for each ae and their contribution to the bride output will cancel. But at the same time, it also follows that an output will only be produced when unequal chanes in strain are produced in the aes. This may be difficult to arrane in some applications and as a result two or three aes in the bride must be replaced with dummy fixed resistances. Then, the strain term for these arms vanishes and the only contribution is the temperature effect. One way to overcome these problems is to use as the dummy aes actual strain aes affixed to a piece of the same kind of metal as that to which the th or active ae is applied. If these dummy aes experience no strain but do experience the same temperature chanes as the active ae (e.., by locatin them in close proximity), they will properly compensate the bride for temperature There are a number of variations in the bride wirin and the confiuration of active and dummy aes that can provide suitable performance for different applications. These are described as follows: Half Bride Confiuration: In this case two dummy resistances are provided in adjacent arms (# and #) as shown and the bride output is then iven from Eq. by: () e e = T T ( ε ε ) E + ( ε ε )E since aes # and # are dummy resistances and therefor experience no strain or temperature chanes (if located toether). DC The bride output is not sensitive to temperature so lon as any temperature chanes occur equally in the two active arms # and # and provided the dummy elements are temperature insensitive. An output is produced only when unequal resistance chanes are produced in aes # and #. The most useful application of this circuit is in the measure of bendin strain in a thin plate or beam. In this case the two aes are mounted opposite each other on opposite (top/bottom) surfaces so that a compressive strain (-ε b ) is introduced in one and an equal tensile strain (ε b ) in the other. Then: AE5 esistance Strain Gae Circuits Pae
e = ( ε ( ε )) E = ε E b b The output is twice that for a sinle active ae and if the two aes are close to each other the temperature chanes will be nearly identical and no temperature sensitivity will be observed. Moreover, if a component of uniform stretchin (or compression) is present in addition to the bendin, its effect would cancel since it would be the same for each ae! Quarter Bride Confiuration: In this case three dummy resistances are provided as shown below and the output is determined from Eq. as: b e e = T ( ε ) E + ( ε )E since now aes #, # and # are dummy resistances and therefor experience no strain or temperature chanes (if located toether). DC For this arranement the output will be temperature sensitive unless the active ae (#) has a zero temperature coefficient. There is no inherent compensation in the bride circuit except for arms #, # and #, #. This circuit is often used when accuracy requirements are not severe (~l0 microstrain) because it requires only a sinle active ae. It also may be the only arranement possible if differential strain measurement cannot be employed. Temperature compensation may be made, however, if resistance # is replaced by a dummy strain ae bonded to the same type of material as the active ae (#) and located as near to it as possible so as to maintain equal temperatures in each. This is essentially a half bride but with only one active ae and is enerally used for hih accuracy ( microstrain) sinle ae measurements. Leadwire Temperature/esistance Compensation When strain ae installations involve runnin sinal lines of more than about 0 feet throuh areas of temperature chane, a sinificant error may be introduced by the resistance of the lines or by the temperature induced chanes in this resistance. To illustrate this point, a 00 foot lenth of two-conductor 6 AWG copper wire is used to connect a sinle strain ae into a quarter-bride circuit as shown above. The wire has a resistance at 68 F (0 C) of 0.8 Ohms/thousand feet, so for the two leads: = 0.8 (00/000) = 8.6 Ohms This is a sinificant resistance compared to the nominal 0 Ohm resistance of a typical strain ae and in addition the wires are also subject to temperature effects! The most common method for overcomin the effects of leadwire resistance is to introduce the lead resistances into the Wheatstone Bride in such a way that their effects do not unbalance the circuit. This is accomplished by arranin the connection of the strain ae to the bride so that one leadwire appears in one arm and the other leadwire appears in an adjacent arm. In this way the initial leadwire resistance appearin in adjacent arms does not unbalance the bride (e.., is not noticed). Furthermore, if the temperature chanes in each leadwire are the same AE5 esistance Strain Gae Circuits Pae
(which can reasonably be assumed for the commonly used twisted shielded pair cables) then the correspondin resistance chane will appear equally in adjacent bride arms and therefore not produce an unbalance. The particular circuitry used for this compensation involves makin a three-wire hookup to the strain ae. An example is shown in Fiure. It should be noted that an apparently superfluous lead wire has been soldered to one of the ae terminals. Closer examination of the circuit should reveal, however, that this junction (A) actually forms one of the four "corner" connections for the Wheatstone Bride. Two of the wires (AB & A E) are part of the bride wirin while the third is either a sinal or power lead. A E A B e D DC C Fiure. Leadwire Compensation Usin a Three-Wire Hookup Provision is enerally made for this type of connection to commercial strain indicatin instruments. It is often referred to as a three-wire hookup. If the leadwires in the above examples are on the order of 00 feet or more, the added resistance in each arm while not unbalancin the bride may amount to a substantial fraction of the ae resistance. If this occurs, the effective ae strain sensitivity is reduced because the same produced by an ε is now a smaller fraction of the arm resistance. In other words: + This situation can be illustrated by considerin the effect of a lead-wire resistance, s, added to for a sinle active arm (quarter-bride circuit) assumed to be initially balanced. wires e S S = total resistance of leadwires to ae DC The bride output can be expressed usin Eq. 6 for a quarter-bride confiuration as: AE5 esistance Strain Gae Circuits Pae 5
e = E = E + S ( + β ) where S β = = leadwire resistance as a fraction of the ae resistance Then usin the definition of Gae Factor from Eq. : e = E = ε E + β + β or * e = ε E = ε E + β where * = ( β ) () + β if it can be assumed that β<<. The result demonstrates that leadwire resistance will have the effect of desensitizin the bride circuit by effectively reducin the ae factor. This suests a compensation scheme. To correct the readins, one can simply use a new ae factor, *, in calculatin strains from voltaes. If commercial measurin equipment is used which indicates the strain directly, then the corrected ae factor, *, should be used rather that the value listed for the strain ae. Note that to find β one must either measure (or compute from wire resistance tables) the leadwire resistance, S. Also, it should be noted that the effect of S is decreased for larer ae resistances (i.e., the effect for a 50 Ohm ae is about / that for a 0 Ohm ae for the same leadwire lenth). Example: We would like to connect a 0 Ohm ae with =.0 to a measurin instrument located 00 feet away. The wire used is 8 AWG copper(8 AWG has a resistance of 6.9 ohms per thousand feet). The corrected ae factor is computed as follows: 6.9 s = 00 = 8.9 Ohms 000 β = 8.9 /0 = 0. *.0 = = =.5 + β. For a 50 Ohm ae the corrected value would be * =.80. BIDGE BALANCING In the previous discussions it has been assumed that the Wheatstone Bride circuit was initially balanced with all resistances equal. In reality, this is not the case due to inherent AE5 esistance Strain Gae Circuits Pae 6
irreularities between even the most accurate of strain aes or dummy resistances. As a result, the bride output voltae, e, is not zero but instead may show an initial unbalance of as much as 0.% of E, (i.e. up to 0 mv for E=0V). This may actually surpass the true strain-induced sinal in many cases. Fortunately; there is a relatively simple and straihtforward method for eliminatin this unbalance without adversely affectin the basic bride circuit. It consists simply of some method for addin or subtractin from the resistance in any one arm so as to satisfy the balance requirement: * = * The adjustment may be made sinly in one arm or relatively (differentially) in two or, more arms. Fiure 5 shows several methods for accomplishin this. In Fi. 5(a) a small resistance is shown added in series with i to increase the arm resistance. The problem here is that this approach can only be used to increase the lefthand side of the above balance equation; thus one must know in advance which arm(s) have the lowest resistance. s s (a) Series (b) Shunt b s b (c) Potentiometric Fiure 5. Strain Gae Balancin Circuits (d) General A second problem is that the added balance resistance must be extremely small in order not to unbalance the bride in the opposite direction. The manitude of this added balance resistance can be determined from the Wheatstone Bride equation. In order to cancel an initial unbalance voltae, a resistance chane must be produced in one arm that is equal to the equivalent straininduced resistance chane in a ae that would produce the same unbalance voltae. Usin Eq. 6 for a sinle active arm (quarter bride confiuration) it follows that: = e 0 E AE5 esistance Strain Gae Circuits Pae 7
In typical situations, the unbalance (output) voltae is a.few millivolts for an excitation voltae of five or ten volts. For example, iven, e o = 0.005 V and E=0 V: or for a 0 Ohm ae: 0.005 = = 0.00 = 0.% 0 = 0.00 0 0. Ohms = This is a very small resistance and is; in fact, comparable in manitude to the contact resistance that develops between matin connectors in a circuit cable. In strain ae brides, this approach is not practical. On the other hand, a much larer resistance can be added in parallel with an arm to reduce its effective resistance and therefore to balance the bride (e.., recall that the resistance, p, of two parallel resistors is iven by p = /( + )). This is shown :in Fi. 5(b). While this solves the second problem above, one must still know in advance which arm has the larest initial resistance. These problems can be remedied by employin the differential arranements shown in Fiures 5(c) and 5(d) where the adjustment is made simultaneously in adjacent arms. Since adjacent arm resistances appear in opposite sides of the balance equation above, this method is capable of handlin either positive or neative initial unbalance voltaes. Aain, however, as with the circuit in Fi, 5(a), the arranement in Fi. 5(c) is not practical because the potentiometer resistance must be extremely small.. Lare values, even a few Ohms, while maintainin a balanced state will effectively increase the nominal arm resistance and thus desensitize the bride. This is exactly the same phenomenon noted in the previous section where the use of lon leadwires added extra (balanced) resistance but desensitized or reduced the bride output for a iven strain. The arranement in Fi. 5(d) offers the most practical and effective way to initially balance a Wheatstone Bride. Several features are noteworthy. The basic structure of the bride is preserved with no balancin components inserted directly into any of the les. This avoids problems with junction contact resistances and desensitization due to added arm resistance. Potentiometer b provides a means for differentially addin parallel resistance to both and. Assumin for the moment that s =0, the maximum balancin effect will occur when the wiper is at an extreme position, in which case b is in parallel with one arm and the other arm ( or ).is shorted. At the midpoint position, b / is in parallel with both and. In order not to desensitize the bride, this parallel combination should not chane the bride arm resistance by more than about %. Usin the parallel resistance formula, one can show that: b i (5) where i is a typical arm resistance and i is the maximum chane allowed. Usin i = % for a 0 Ohm ae yields b = Ohms. esistance s is included in the circuit simply to limit the maximum balance action which occurs at the extreme potentiometer settins. In this case, s is in parallel with one arm and b is simply connected between opposite bride corners. Typical values for s rane between and 5 kohms. The particular arranement of s and b within the bride is unimportant; b can be connected between sinal corners or excitation corners with no difference in performance. For best stability and minimum drift, these resistances should be i AE5 esistance Strain Gae Circuits Pae 8
metal film or wirewound desins (NOT carbon composition) with a temperature coefficient of less than 5 ppm/f. CALIBATION The output from a strain ae bride is proportional to chanes in resistance of all of the arms. In most situations, only one or two arms are active and it is desirable to be able to provide some means of assurance that the circuit is workin properly. The Wheatstone Bride circuit is ideally suited for this purpose because it is relatively easy to affect a chane in resistance in one or more arms that is proportional to a known physical parameter. In order to chane the resistance in one arm, two approaches are possible as shown in Fi. 6. A resistance may be added in series to increase the arm resistance or else a resistance may be added in parallel to reduce the arm resistance. Of these two, the parallel or shunt connection is preferred because it requires the least modification of the bride and utilizes a more practical resistance value. The series connection requires very hih quality switch contacts in order to maintain bride continuity and thus preserve accuracy. It also requires use of very small calibration resistances which are not easily fabricated. On the other hand, the parallel connection requires a relatively lare resistance and the switch contacts are not a direct (internal) part of the bride circuit. c c (a) Series (b) Shunt Fiure 6. Strain Gae Bride Calibration Circuits An analysis of the calibration circuit in Fi. 6(b) can easily be carried out by use of the basic bride equation (Eq. 6) and the parallel resistance equation. The new or altered arm resistance in Fi. 6(b) is iven as: and thus: = c + c = = = + c c c + c = + c The chane in resistance is equivalent to an imainary strain whose manitude can be computed from the equation definin the ae factor (Eq. ) as: AE5 esistance Strain Gae Circuits Pae 9
/ ε = = (6) + c Normally, this is solved to yield the value of c required to produce an equivalent strain output: c = ε (7) ε where the simplification is based on the fact that ε is a very small value (microstrain). Example: Suppose it is desired to connect a shunt calibration resistance across a 0 Ohm ae with =.09 to simulate a 000 microstrain readin. The value of c is then iven by Eq. 7 as: c = 0 = 8, 588 Ohms.09 0.00 This particular value of resistance is not commonly available. Suppose instead that a value for c is selected in advance to be c =0 kohms. Now, from Eq. 6 the indicated strain will be: ε = + c =,85 microstrain As a final note, it should be pointed out that at least one strain ae manufacturer makes precision shunt calibration resistors especially for this purpose. The resistances are available for specific combinations of ae resistances, values, and apparent or induced calibration strains. Strain Gae Switchin Circuits Strain aes are almost always used in multiples since a common application is to measure the strain at various locations on a part or within a system. It is impractical to use separate strain ae indicator circuits for each strain ae used in such situations, and as a result, some kind of switchin circuitry is enerally employed. And dependin on whether the application is for quarter bride, half bride or full bride ae wirin, the switchin circuitry be somewhat different. The followin sections discuss some of the alternatives and make note of special problems to anticipate. Interbride Switchin The simplest approach is interbride switchin which involves switchin complete full bride circuits as shown in Fiure 7. The most common approach is to supply a common excitation voltae to all of the brides concurrently and then switch the outputs to the strain indicator usin a two-pole, multi-position switch. No switchin in done within the Wheatstone Bride itself and the sinal switchin involves neliably small currents, the demands on the switch are minimal and modest quality switches may be used without difficulty. If the power is also switched, there will be modest current flowin in the bride and the switch must be desined accordinly. Switchin the power supply is normally only done if the aes must be operated for short periods just to take a readin (this may be done to minimize thermal drift from resistance heatin in the aes if very hih voltaes are used to et the hihest possible sensitivities). The key advantae for interbride switchin is that it maintains the full interity of AE5 esistance Strain Gae Circuits Pae 0
the Wheatstone Bride and can also include balancin circuitry for each switched bride. Of course the disadvantae is the need to work entirely with full brides which is often difficult to do. If quarter bride confiurations are used (typically for individual strain measurements), then this interbride switchin circuit will require use of dummy aes for each sinle active strain ae and this can be costly and complex. DC Bride # Bride # Bride # Added Brides eadout e Fiure 7. Strain Gae Interbride Switchin Intrabride Switchin Quarter Bride Intrabride switchin involves switchin some of the arms within the bride while maintainin only a sinle Wheatstone Bride circuit. This is the approach most commonly used when workin with quarter bride confiurations. Fiure 8a shows a typical quarter bride confiuration in which a sinle pole, multiposition switch is used to complete the fourth arm of the Wheatstone Bride that contains the active strain ae. This is perhaps the simplest of all the switchin confiurations in terms of the number of components but it introduces the followin problems: The added resistance of the switch appears alon with the ae resistance for the switched arm and this may be a substantial factor. Switch resistance can also vary considerably between switch operations and this adds a precision uncertainty. Hih quality switches with old plated wipin contacts can often overcome these problems but at a cost. There is no easy way to balance out any residual strain in individual switched aes without addin additional switch poles. Intrabride Switchin Half Bride Intrabride switchin can also involve switchin two adjoinin arms of the Wheatstone Bride (half-bride switchin) as shown in Fiure 8b. In this confiuration, two arms (A, B) form one half of the bride and are common for all of the switched aes. In this particular confiuration, all switched aes are powered continuously and only the bride outputs are switched (the unswitched half of the bride simply defines the reference for the bride output circuit). Balancin can also be added to each of the external half brides usin a circuit similar to that shown in Fiure 5d. This bride confiuration provides perhaps the best versatility and overall performance. If quarter bride (sinle active ae) confiurations must be used, this circuit will require addition of a sinle dummy ae for each active ae. AE5 esistance Strain Gae Circuits Pae
Switch DC Half Bride Active Gaes DC Quarter Bride Active Gaes e e (a) Quarter Bride (a) Half Bride Fiure 8. Strain Gae Intrabride Switchin Confiurations Commercial Switchin and Balancin Units When commercial strain indicators are used to manually measure multiple strain aes, a matchin commerical switchin and balancin unit (SBU) is often employed. These SBU s typically contain provisions for full, half and quarter bride switchin and include separate balancin controls for each channel. For the hihest performance and accuracy, the switches and connectors in these SBU s are usually old plated and desined to minimize spurious contact resistance. The wirin between the SBU and the companion strain indication is usually indicated on the instruments and may seem to bear little resemblence to the circuitry described in these notes. Nonetheless, with a little persistence in studyin the schematic diarams, it should become obvious that simple variations of the same circuitry are involved in all cases! Strain Gae Applications One of the most appealin characteristics of strain aes is their versatility, particularly when incorporated into a Wheatstone Bride. On first exposure, the complexity of the Wheatstone Bride circuit and the need to provide either active or dummy aes in all arms seems almost overwhelmin. It seems downriht inefficient to have to provide additional bride arms just to be able to sense the strain in a sinle active strain ae! However, on closer inspection and after more extensive experimental work, the versatility of the bride may become more evident. For example, while it is often true that only a sinle strain ae measurement is needed, it is usually also true that the ae is expected to respond to a particular type of loadin the desiner of the experiment may have one to reat lenth to assure this. On the other hand, it may be possible to install two strain aes in such a way that the undesired response is automatically rejected by the bride, even thouh it has been sensed by the aes themselves! This is particularly important when makin measurements of D strain or stress states where a rosette ae can be used very effectively to measure the shear strain which cannot even be directly detected by the strain aes themselves. In this section, a number of different strain ae applications will be described to further illustrate the versatility of these devices and to reinforce the basic and underlyin strenth of materials theory. Perhaps other applications will suest themselves. AE5 esistance Strain Gae Circuits Pae
Bride Equation The basic Wheatstone Bride strain ae circuit can be described by the simple relation developed in Eq. (7): e = ( ε ε + ε ε ) E (7-repeated) where =ae factor, E=bride excitation voltae and ε I are the strains sensed by the individual bride aes. This relation is accurate for most common situations but it does not apply to the followin cases: (a) lare strains (in excess of 0,000 microstrain), (b) aes with different values for in different arms, (d) brides in which the sensin device is current sensitive rather than voltae sensitive. It is doubtful that any of these limitations will pertain to most applications but they are stated for completeness. One of the most useful aspects of Eq. 7 is that the strains are additive with alternatin sins as one moves around the bride. As a result, it is possible to deduce these important bride rules : equal chanes in strain in adjacent arms produces no bride output, equal chanes in strain in opposite arms produces bride output at double the sensitivity of a sinle active arm, opposite chanes in strain in adjacent arms produces bride output at double the sensitivity of a sinle active arm, opposite chanes in strain in opposite arms produces no bride output. equal chanes in strain in all arms produces no bride output. These relatively simple rules are easily confirmed from Eq. and are the source for the different measurement confiurations discussed below. In order to apply these rules and to correctly assess the results, it will be useful to define a nomenclature as follows. Assume that the reference bride performance is that associated with a quarter-bride circuit, that is a bride with only a sinle active ae. In this case Eq. 7 indicates that the output will be: e = E ε To reflect the chanes in bride performance by addin one or more additional active arms, a bride factor, K, will be added so that the new bride equation is: e = K E ε () For example, a quarter-bride has a value, K=, while a bride in which all arms contribute equally and additively, K=. Practical values lie somewhere in between. Sensin Strain in a Member Under Uniaxial Load It is often necessary to measure the strain in a prismatic member (a bar) subjected to an axial load. A common example miht be a truss element which is desined to carry axial load. In this case application of a sinle strain ae oriented in the axial direction on the bar would appear to be sufficient. However, several problems arise. First, there is the problem of what to do about the other arms of the Wheatstone Bride, but second, it is not always so easy to assume that a sinle ae will correctly sense the axial strain in the bar. For example, while the stress state AE5 esistance Strain Gae Circuits Pae
may be uniaxial (consist of only a sinle nonzero stress), the strain state is not, and there are sinificant lateral strains that the ae miht sense, especially if it is not accurately alined. Moreover, any sliht bendin in the member (due to initial eccentricities, for example) or other irreularities miht cause the axial strain to vary across the cross section of the bar. Table 5 below summarizes some of the most common situations that miht be encountered and provides wirin notes and K values. Table 5. Bride Confiurations for Uniaxial Members No. K Confiuration Notes A- Must use dummy ae in an adjacent arm ( or ) to achieve temperature compensation A- ejects bendin strain but not temperature compensated; must add dummy aes in arms & to compensate for temperature. A- (+ν) A- (+ν) Temperature compensated but sensitive to bendin strains Best: compensates for temperature and rejects bendin strain. Sensin Strain in a Member Under Bendin Load Quite the opposite to what was considered in the previous section can also be true. That is it may be necessary to sense the bendin induced strains in a prismatic member and NOT the strains due to axial loadin. This is usually the case when dealin with beams or other so-called flexural elements. For these cases, the bride can be wired so that the equal and opposite strains that are induced on the upper and lower surfaces of a simple beam will appear in adjacent arms where the strains will be combined. Even when the beam cross section is such that the centroid is not equal distances from the top and bottom surfaces (e.., a tee section), the strains will still be of opposite sin and will add constructively in the bride equation. Table 6 below summarizes the most common confiurations for flexural applications. Table 6. Bride Confiurations for Flexural Members No. K Confiuration Notes F- Also responds equally to axial strains; must use dummy ae in an adjacent arm ( or ) to achieve temperature compensation AE5 esistance Strain Gae Circuits Pae