Wheatstone bridge

STRAIN GAUGECIRCUITRY:

WHEATSTONE BRIDGE

Instruments for Engineering Measurement29 Van Riebeeck StreetGerdviewGERMISTON 1401South Africa

P.O.Box 2222ZA-1416 PRIMROSE

Tel: (011) 828-6169Fax: (011) 822-1377

Compiled by Louis Eder / Instruments for Engineering Measurement

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Foreword (a word from our sponsor…)

Instruments for Engineering Measurement was established in1992, at first as a “systems integrator” and an importer of highquality loadcells from – the previously unknown – KaliberInstrument and Measurement Technology Company in Budapest,Hungary. Because there are a great number of loadcellmanufacturers in the world, each offering better and better loadcellsat lower and lower prices, our Customers were limited to those whoneeded something special, or at least a loadcell which could enduresevere dynamic loading and adverse environments without beingdamaged.

These Customers were few and far between, and our attentionturned to another of Kaliber’s products: the wire grid strain gauge.This type of strain gauge was manufactured a long time ago byPhilips, TML, HBM, and possibly others. The available literature (T.Potma: Strain Gauges) refers to wire grid strain gauges on a paperbacking, and the great difficulties in obtaining accurate readingswith these. Problems were experienced with production, especiallywith attaching terminals to the grid, without contaminating the paperbacking with solder fluxes. Relatively high cost of production, a highreject rate, coupled with the lack of modern-day adhesives, and aspecial “feel” needed to apply these strain gauges propagated theidea that strain gauge application was a “black art” known only to aselect few. The select few, in turn, capitalised on this idea, and keptthe experience to themselves.

With the development of printed circuit techniques, strain gaugesrapidly became a mass produced item, with all kinds of grid shapesand sizes imaginable, which could be made with the photoresistand etching method. Ultra-small strain gauges were feasible,especially for applications in loadcells and other transducers, wherethe measuring element is necessarily small, requiring miniaturestrain gauges with grid lengths of 3 mm and less to measure thestrain on the small load bearing element. In stress analysis too,many tiny strain gauges could be made on one small backing tomeasure strain gradients on small parts.In the early 80’s, HBM stopped the manufacture of wire grid straingauges, and TML continued with them until the early 1990’s, thenstopped also.

The wire grid strain gauges of Kaliber were of a much moresound design than the paper-backed type. Phenolic resin was usedas a backing, and constantan wire of 20 - 30 microns diameterused for the grid. The manufacturing process allows the grid to becompletely embedded in the backing, which is not more than 50 -100 microns thick, and heavy duty copper solder terminals to beintegrated into the strain gauges. Of course, this allows easyhandling, and attaching of measuring cable directly to the straingauge. Our first experiences with these strain gauges was that theywere easy to apply and extremely stable, especially at strain levelsbelow about 500 µm/m. Their relatively long grid (10 and 20 mm)allows low error measurement, with excellent long term stability.Soon we discovered that customers who were in the market forstrain measurement were not interested in learning the applicationprocess, rather they were happy to give us an order to do thecomplete job. We accepted the challenge, and did not invoice untilthe job worked to the customer’s satisfaction. In this way, we didour first strain gauge job, instrumenting of a 100t garage typepress, in Welkom. This was rapidly followed by the strain gaugingof the big 750 t press at the same place, then a 100t Amsler press.The presses were calibrated by SABS, and were found to be wellwithin the accuracy requirements. In the case of the 750 t press,the overall linearity was better than 0,2%, much better than anyother measuring system used on that press. Kaliber themselves didnot believe that their product was that good ! But, the long termtemperature stability was not so good, and the zero point had to beadjusted at the beginning of each test.

Soon, we started manufacturing our own amplifiers, becausewhat was available in the market was either good and frighteningly

expensive, or affordable but unsuitable. To date, we have deliveredsome very fine systems, the latest with microcontroller technology,more than 16-bit resolution and computer interface. We stay awayfrom carrier frequency amplifiers, because of cable capacitanceproblems, and DC amplifiers have come a long way in long termstability. Good amplifier design is of paramount importance, and itis possible to produce systems at a reasonable price which are ahappy marriage of old-fashioned analogue technology, and moderndigital processors, rather than try to eliminate analogue problemswith software !

In November 1997, we purchased the entire wire grid straingauge manufacturing process from Kaliber, who decided to give upthe wire grid strain gauges, and to start making foil grid straingauges for their own loadcell and transducer production. Since westarted with the wire grid strain gauge manufacture, we have madea few improvements to the form of the strain gauge. The copperterminals are larger and do not extend to the edge of the gauge.Also, for half and full bridge measurement, we offer matched straingauge pairs, whose elements are chosen from the same batch,even from the same sheet that they were made. The huge benefitof this was observed at an outdoor site near Stilfontein, where theambient temperature varies greatly. With a temperature variation atthe metal surface of +30 to +13 °C, (just before sunset to justbefore sunrise) we observed a measured value which varied by 1count, with a measuring range of 0,2 mv/v representing 327 countson the digital display. This, with no temperature compensation atall, and with a constant mass applied, therefore the measuredvalue contained the span temperature error as well as the zeropoint temperature error. To save installation time, we pre-wire thestrain gauges as half bridges with external terminals, and weactually did an experiment to check our installation speed. Sandingand degreasing of the surfaces, sticking 16 gauges in 8 places,wiring up, insulation testing, sealing, running the cable andinstalling the amplifier, connecting it to 220v power took 1 hour and20 minutes for 2 people.

We have found that the wire grid strain gauge offers somebenefits: the cross-section of the grid is circular, therefore thegauge factor is very predictable, being within 1% of the theoreticalvalue of 2,05. The temperature coefficient of the gauge factorhappens to inversely match the temperature coefficient of theelasticity of steel, within a few parts per thousand. The temperaturecoefficient of the zero point is zero, which means that we can notproduce “temperature compensated” strain gauges. Rather, wesuggest to use a matched pair in a half bridge, and then the zeropoint temperature error cancels completely. The applicationprocess that we use, together with their large and robustconstruction makes it possible to apply these strain gauges even industy, windy conditions, and get measuring results as good aswhen applied in a clean laboratory. We have discovered that readilyavailable application materials produce excellent results. Ourcovering material, first applied in 1997, is re-enterable, yet protectsthe strain gauges for years in mint condition, despite ourcompetitors’ comments that this type of application is strictly short-term…

In recent months, our strain gauges have been exported toEurope, Australia, and Israel, each time with accuracy which wasabove the client’s expectation. Although our experience shows thatthis is a precise and predictable technology (and not a black art),there are some applications where we find temperature gradients,and these can cause headaches with regard to temperature error.Besides applying temperature compensation as the loadcellmanufacturers do, simple prevention such as thermal insulation isan effective solution. As wonderful as the strain gauge technologymay be, some applications demand extra attention, and the lessonto be learned is that we never stop learning. Each project is athrilling, challenging and learning experience.

Louis Eder


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CONTENTS

GAUGE FACTOR ...................................................................................................................................... 4quarter bridge. .......................................................................................................................................... 6half bridge. ................................................................................................................................................ 9

cantilever beam, ...................................................................................................... 12

full bridge ................................................................................................................. 12

Resistivity of copper:................................................................................................ 15

Temperature coefficient of resistance of copper wire: ............................................. 17

AMPLIFIER INPUT RESISTANCE .......................................................................................................... 22USE OF DUMMY GAUGES FOR TEMPERATURE COMPENSATION ................................................. 25THREE WIRE CONNECTION ................................................................................................................. 27FOUR WIRE CONNECTION ................................................................................................................... 28METHODS OF COMBINING READINGS FROM MULTIPLE BRIDGES............................................... 29

PARALLEL CONNECTION OF THE BRIDGES ...................................................... 30

SERIES CONNECTION: ......................................................................................... 31

FORMULAE ............................................................................................................................................. 331. QUARTER BRIDGE ..................................................................................................................... 332. HALF BRIDGE.............................................................................................................................. 333. FULL BRIDGE.............................................................................................................................. 34


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GAUGE FACTORThe change of resistance caused by strain in metallic strain gauges is determined bytheir gauge factor. The gauge factor (k) is the most important characteristic of thestrain gauge and is about the same for all available strain gauge types from thevarious manufacturers. The gauge factor is the constant that determines the electricalsignal as a function of mechanical strain:

The value of the gauge factor k is around 2,which means that in the usual type ofmeasurement on metals, with a strain of 1000µε using 120 Ω strain gauges results in aresistance change of 0,24 Ω.

To measure such a small resistance change accurately precludes the use ofinstrumentation such as ohmmeters. A more precise method is to use comparisontechniques, using one or more known, stable, fixed resistors:


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Initially, R1 = R2 and the value of UOUTwill be:

UEXC. R2UOUT = -------------

R1 + R2

UEXCUOUT = -------

2

= 0,5 UEXC

A change of resistance, induced by strain, of 0,24 Ω will cause an output voltage of:

UEXC. 120UOUT = -----------------

120,24 + 120

120. UEXCUOUT = -----------

240,24

UOUT = 0,4995 . UEXC

It is quite difficult to measure a change between 0,5.UEXC and 0,4995.UEXC with anydegree of accuracy.

Two possible ways of overcoming this difficulty are:


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1. Generate equal and opposite UEXC voltages and reference UOUT to zero.

This arrangement demands very stable power supplies,as even a very small common-mode swing of + and -UEXC will result in an apparent, erroneous output.

2. Create an artificial zero point, using two fixed resistors:

The matching of + and - UEXC nowbecomes unimportant, as the outputdepends only on the potential differencebetween the junctions of R1 and R2, andR3 and R4.

This is a widely used circuit for themeasurement of an unknown resistor, andis known as the Wheatstone bridge.

In strain gauge work, we refer to thiscircuit, which has only one of the fourelements actually varying, as a

quarter bridge.


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HISTORICAL NOTE:

There are strong advantages to using aWheatstone bridge for strain gaugecircuits. The bridge allows addition andsubtraction of strain signals which enablesmagnification of wanted signals andcancellation of interferences as we shallsee later.

Our output signal is:

UOUT = UA - UB

UEXC . R1 UEXC . R3 = ----------- - -----------

R1 + R2 R3 + R4

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

Using the numerical example of ∆R = 0,24 Ω:

Copyright 1995 by Grolier Electronic Publishing, Inc.Wheatstone, Sir Charles

Sir Charles Wheatstone, b. Feb. 6, 1802, d. Oct. 19, 1875, was an Englishphysicist and inventor whose work was instrumental in the development of thetelegraph in Great Britain. Wheatstone served (1823-34) an apprenticeship asa musical-instrument maker. His work in acoustics won him (1834) aprofessorship of experimental physics at King's College, London, where hispioneering experiments in electricity included measuring the speed ofelectricity, devising an improved dynamo, and inventing two new devices tomeasure and regulate electrical resistance and current: the RHEOSTAT andthe Wheatstone bridge. He worked on magnoelectricity and submarinetelegraphy, and he suggested the STEREOSCOPE. In 1837 he designed,with William F. Cooke, an electric telegraph system that became standard inBritain in 1840.

Bibliography: Bowers, Brian, Sir Charles Wheatstone FRS, 1802-1875 (1975)Hubbard, Geoffrey, Cooke and Wheatstone and the Invention of the ElectricTelegraph (1965).


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UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120,24 120= ----------- - -------

240,24 240

= 0,500 499 5 - 0,5

= 0,000 499 5 V / V

or 0,4995 mV / V

Let us examine the following case where the strain is compressive, and the resulting∆R negative:

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

119,76 120= ---------- - ------

239,76 240

= 0,499 499 4 - 0,5

= -0,000 500 6 V / V

or -0,5006 mV / V

Here we see evidence of the inherent non-linearity of the Wheatstone bridge whenused for quarter bridge strain measurement. ERROR% refers to the differencebetween tensile and compressive strains of equal magnitude.


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Applying R2 in addition to R1 and makinguse of the property of the strained memberknown as Poisson's ratio adds an extraactive element to the quarter bridge, makingit a

half bridge.

The reason why this is possible is that whileR1 stretches, R2 will simultaneouslycontract. (The contraction of R2 will,however be smaller than the stretch of R1 -in this case 30%.)


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Using the previous strain value of 1000 µm/m and initial resistances of 120 Ω, wehave:

∆R1 = +0,25 Ω and

∆R2 = 0,3 . -0,25 Ω

∆R2 = - 0,075 Ω

UOUT R1 R3------ = --------- - ---------UEXC R1 + R2 R3 + R4

120,25 120= -------- - ----- 239,925 240

= 0,5011982 - 0,5

= 1,1982 mV / V

Beside the increased signal, let uslook at the linearity error of thiscircuit.


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Applying compression to create compressive strain, the equation becomes:

∆R1 = -0,25 Ω and

∆R2 = 0,3 . +0,25 Ω

∆R2 = + 0,075 Ω

UOUT R1 R3------ = --------- - ---------UEXC R1 + R2 R3 + R4

119,75 120= -------- - ----- 240,075 240

= 0,4988024 - 0,5

= - 1,1976 mV / V

Once again, the ERROR reflected in the graph shows the difference between thesignals of two strains of the same magnitude and opposite polarity.

Note the convention of expressing the output of the Wheatstone bridge as a voltageratio. Because the bridge itself does not generate any current, its output will always bea fraction of the excitation voltage. Increasing the excitation excessively is not possiblebecause of self-heating of the strain gauges, whose current should be limited to lessthan 25 mA generally. For 120 Ω gauges, this means a maximum of 6v excitation, andan output at 1000 µε of 3 mV.


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In measuring certain types of strain, we can take advantage of the symmetricalproperties of the strainon the measuredobject.

Ideally, there will be twopoints in equal comp-ression and two pointsin equal tension, themagnitudes of tensionand compression alsobeing equal.

Such a case is thecantilever beam,

where, by careful placement of the strain gauges, we can obtain the requiredsymmetry.

Connecting the strain gauges as shown, willproduce a

full bridgewhose properties we will now examine.

Taking the original resistance and strainvalues, we get:

R1 = R2 = R3 = R4 = 120 Ω

∆R1=∆R2=∆R3=∆R4 = 0,24 Ω:

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120 + 0,24 120-0,24= -------------------------- - ------------------------

120+0,24+120-0,24 120+0,24+120-0,24

120,24 119,76= ----------- - ----------


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240 240

= 0,501 - 0,499

= 0,002 V / V

or 2,0000 mV / V

We can see that reversing polarity of the strain will simply swop around the two terms,but the output will be of the same value (and opposite polarity).

One important property of the fully active full bridge is therefore:

NO LINEARITY ERROR.

Some applications where this type of measurement can be utilised are:

bending (as illustrated above)torsionshear.

In each case, two pairs of strain gauges can be used, each pair being subjected toequal and opposite strains.

Let us now examine the case where the cantilever beam is strained in a direction thatwe do not want to measure:

The four strain gauges areall compressed, that is,their values are reduced.Looking once more at theprevious schematicdiagram and using thebridge formula, we have:


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R1 = R2 = R3 = R4 = 120 Ω

∆R1=∆R2=∆R3=∆R4 = 0,24 Ω:

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120 + 0,24 120+0,24 = -------------------------- - ------------------------ 120+0,24+120+0,24 120+0,24+120+0,24

120,24 120,24 = --------- - ----------

240,48 240,48

= 0,5 - 0,5

Another important property of the full bridge is:

CANCELLATION OF UNWANTED STRAINS

Temperature variations in the measured member can cause unwanted strainreadings. We can make use of the fact that the measured member changesdimensions equally in all directions in response to temperature change, but a word ofcaution when measuring cylindrical or other irregular members ! (The dimensionchange is of the metal itself, not taking into account the thickness of adhesive and gridcarrier. This can cause some considerable errors.)

Now that we have considered quarter, half and full bridge circuits, let us examine afew practical applications where usage of bridge circuits helps to solve problems...


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Here, we have to measure a strain value at a particular point on a structure.

On this structure, we have no means of using the full, or even the half bridge, only oneactive strain gauge is used. The other three-quarters of the bridge is in the measuringinstrument.

We have added extra resistance to the active quarter bridge, in the form of connectingwire. The 0,22 mm2 wire is commonly available, and its resistance is as follows:

Resistivity of copper:

ρρρρ = 0,017 ΩΩΩΩ / m / mm2 at 0°C

and our wire resistance is therefore:

0,017 . 10RL1 = -------------

0,22

= 0,7727 Ω (if the temperature is 0°C)

The return wire has the same resistance, so we have a total extra resistance of0,7727 . 2

= 1,545 Ω


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This results in the following equivalent circuit:

Applying the Wheatstone bridge formula:

R1 = 120 + 1,545 Ω (strain gauge plus leads)

R2 = R3 = R4 = 120 Ω

∆R1= 0,24 Ω (our initial scale of 1000 µε)

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120 + 1,545 + 0,24 120 = -------------------------- - ------------ 120+0,24+120 120+120

121,785 = --------- - 0,5

240,24

= 6,9305 mV / V


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Knowing about non-linearity in the quarter bridge circuit, we should also check whatthe instrument reading will be with zero strain:

R1 = 120 + 1,545 Ω (strain gauge plus leads)

R2 = R3 = R4 = 120 Ω

∆R1= 0 Ω (zero strain)

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120 + 1,545 120 = -------------------------- - ------------ 120+120 120+120

121,545 = --------- - 0,5

240

= 6,4375 mV / V

The signal span will therefore be :

0,493 mV / V for 1000µε , and not the 0,4995 mV / V for 1000µε that we expect.

This is an error of some 1,3 %, and should be taken into consideration.

Of this signal, 0,493 mv/v is due to the strain of 1000 µε, and the rest, being 6,4375mv/v, is due to lead resistance. This is a large value, but strain amplifiers will havesufficient adjustment to balance this value to zero.

Another, more tricky problem is the fact that copper, in spite of being one of the bestconductors of electric current, has quite a large temperature coefficient of resistance

Temperature coefficient of resistance of copper wire:

αααα r(Cu) = 0,004 ΩΩΩΩ / ΩΩΩΩ / °C

With our wire resistance of 1,545 Ω, and an expected ambient temperature range of(say) 0°C to 25 °C (typical Highveld winter's day in a factory), the wire resistancevalues will be:


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at 0°C: 1,545 Ω

at 25°C: 1,545 Ω +(1,545 . 0,004 . 25)

= 1, 6995 Ω

Because our amplifier cannot tell the difference between strain induced resistancechange and temperature induced resistance change, it will measure a strain causedby a change in resistance of:

1,6995 - 1,545 Ω = 0,1545 Ω

which will correspond to a strain reading of:

∆R---- = k. ε R

∆Rε = ----

R.k

0,145= --------------

121,545 . 2

= 596,4 µε

for the ambient temperature change alone ! (Assuming, of course, that thestrain gauge itself is fully compensated to the metal of the member being measured,and gives no output change for the member's own temperature expansion.)

Using the Wheatstone bridge's properties, part of the problem can be compensated. Itis simply a matter of where we connect the measurement point:


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We actually add more resistance to the circuit, but, please note that, because there isno current flow to the amplifier, the new resistance will have no effect on themeasurement.

Note also, that one of the lead resistances, RL2, now belongs to R2, not to R1 anymore .

Let us examine the initial conditions once again:

τAMB = 0°C , ε = 0, RL1=RL2=0,7727 Ω

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120,7727 120 = -------------------------- - ------------ 120,7727 + 120,7727 120+120

= 0 mv/v


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τAMB = 25°C, ε = 0, RL1=RL2=0,7727 Ω +(0,7727 . 0,004 . 25) = 0,85 Ω

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120,85 120 = -------------------------- - ------------ 120,85 + 120,85 120+120

= 0 mv/v

We have got rid of the temperature problem, and our amplifier no longer has tobalance out a large offset value, but we should also look at the error in the measuredstrain value:

τAMB = 0°C , ε = 1000µε, RL1=RL2=0,7727 Ω, ∆R1 = 0,24 Ω

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120 + 0,7727 + 0,24 120 = ------------------------------------- - ------------ 120,7727 + 0,24 + 120,7727 120+120

121,0127 120 = ------------- - ----- 241,7854 240

= 0,4963 mv/v

τAMB = 25°C, ε = 1000µε, RL1=RL2=0,85 Ω, ∆R1 = 0,24 Ω

UOUT R1 R3------ = ----------- - -----------UEXC R1 + R2 R3 + R4

120 + 0,85 + 0,24 120 = ------------------------------ - ------------ 120,85 + 0,24 + 120,85 120+120

121,09 120 = ------------- - ----- 241,94 240

= 0,4959 mv/v

The temperature change produces a span error of 0,08% which can probably beignored.


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A very useful strain gauge arrangement for normal strain measurement is shown here.It encompasses complete temperature compensation (in the accuracy range thatcan be considered acceptable), and produces minimal linearity error and thebiggest signal that can be had under the circumstances.

It uses two strain gauges applied in the main strain direction, and two at right angles,making use of the Poisson effect.

UOUT R - ∆R1 R + ∆R3------ = -------------------------- - -------------------------UEXC R - ∆R1 + R + ∆R2 R - ∆R3 + R + ∆R4

∆R1= R.k.ε ∆R2= ν.R.k.ε ∆R3= ν.R.k.ε ∆R4= R.k.ε

Taking a value of R=120, k=2, ν=0,3 and values of ε from -1000 to + 1000 µm/m, weget the following:

C

Quite an acceptable performance, with the additional feature of cancelling anysuperimposed bending (although this is not necessarily desirable).

AMPLIFIER INPUT RESISTANCE

Having examined various bridge circuits, it would have been noted that in each case,stress was placed on the fact that no current may be drawn from the outputconnections, otherwise the bridge circuit will not function correctly. This imposescertain requirements on the input circuitry on the measuring amplifier, regarding inputresistance. At the same time, we should also consider that (especially in the earlydays of strain measurement), the only instruments that were available for measuringthe output of a bridge were moving-coil galvanometers, which had a very low inputresistance.

Copyright 1995 by Grolier Electronic Publishing, Inc.galvanometer

gal-vuhn-ah'-muh-tur

A galvanometer is an instrument that measures the amount of electrical current by converting electrical energy into thephysical displacement of a coil, which in turn moves a pointer or light beam. The device was named for Luigi GALVANI, an18th-century Italian physiologist and physicist.

In a galvanometer, a coil of fine wire is suspended between the poles of a permanent magnet, so that when the coil ismagnetized as current passes through it, the like poles of magnet and coil repel each other and cause an attached pointer todeflect across a calibrated scale. When a light beam is used instead of a pointer, a mirror is mounted on the side of themoving coil and a fixed beam of light is directed at the mirror. As the coil turns, the reflected image of the light moves along atranslucent, calibrated panel. The coil may be mounted on a spindle, whose ends turn on rubies or very hard steel.

The direct-current ammeter is a type of calibrated galvanometer that measures larger currents a calibrated galvanometermay also be used as a direct-current voltmeter, which measures direct voltage using Ohm's law. Galvanometers arecurrently being replaced by modern digital instruments.

Leslie W. Lee

Bibliography: Cooper, William D., Electronic Instrumentation and Measurement Techniques (1978).

ompiled by Louis Eder / Instruments for Engineering Measurement

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The earliest measuring instruments for strain were called bridges or compensatorsand they used no electronic amplifiers at all. Their operating principle was based onprecision switchable resistors, which were adjusted until the bridge output was zero,indicated by a galvanometer. When this condition occurs, no current flows betweenthe output terminals of the bridge, and no error is introduced, even if thegalvanometer's resistance is very low.

The schematic of such an instrument is shown here in rough form:

For quarter bridge operation, R3 and R4 arefixed. R2 comprises precision resistors selectedby a multi-way rotary switch and also acontinuously variable FINE adjustment, eachwith calibrated dials.

The strain gauge wiring criteria are the same aswe have previously discussed, that is, attentionmust be paid to lead resistances.

At the start of the measurement, the R2 dialsare set until the galvanometer reads zero. Anote is made of the dial settings, and this is thezero condition.

With the strain applied, the dials are set oncemore to produce a zero on the galvanometer.The difference between the "zero" and"strained" settings on the dials produced therequired measured value.

The actual "bridge" instrument was quite refined. Dial calibration was directly in µm/m,and there was also provision for gauge factor adjustment. One manufacturer of such aBridge was the Swiss HUGGENBERGER, and there is at least one such instrumentstill in (semi)active use in South Africa.

In fact, the highest precision strain meters (until recently) still employed the"compensator" or null balance technique, as it is the most precise, offering inherentlinearity compensation and independence of measuring amplifier input current.

Let us examine the influence of the current drawn from the Wheatstone bridge onaccuracy.


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For the purpose of this discussion, it is convenient to replace the conventionalWheatstone bridge with an equivalent circuit:

UGI = -------- RB + Ri

RB is the bridge resistance, with RB = R1 = R2 = R3 = R4, and Ri is the inputresistance of the amplifier, acting as shown.

The voltage appearing at the input of the amplifier, by Ohm's Law, will be:

URi = I. Ri

Ri = UG . --------------

RB + Ri

As Ri tends to infinity, so URi tends to UG , because the fraction will tend to 1.


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The span error introduced by the amplifier's input resistance will be:

Ri = 100 . ( 1 - -------------- ) %

RB + Ri

As users of strain gauge amplifiers, we expect that this instrument will provide us withan acceptably accurate measured value. The graph shows the span error introducedby amplifier resistance which is less than infinite. An especially irritating factor ishaving an altered apparent strain reading when using strain gauges with a differentresistance.

USE OF DUMMY GAUGES FOR TEMPERATURE COMPENSATION

Although many modern strain gauges incorporate self-temperature-compensation, itmay be wise to examine the manufacturer's specifications in detail to establish exactlyto what extent this compensation is applicable to the job at hand.

In many applications, it may be better not to rely on the self-compensation, but to usehalf- or full bridge circuits wherever possible. In that case, the self-temperaturecompensating properties of the strain gauges become of no consequence. If the fullbridge is not feasible, a half-bridge circuit offers a good compromise, and if even thisis not practical, we need to look at the possibility of temperature compensating"dummy" strain gauges.


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The compensating or "dummy" gauge R2 is cemented to a piece of metal which is thesame metal as the mainmember, to ensure that itstemperature coefficient ofexpansion is the same. Thedummy piece of metal isattached to the main memberin such a way that it is freefrom stress, but is exposed tothe same ambienttemperature. An importantpoint to note is that if thedummy metal is substantiallysmaller than the mainmember, then there may betransient temperature effects,if the ambient temperatureshould change rapidly, forexample if exposed tosunlight or other radiantsources. The smaller piece ofmetal will heat more rapidly,and produce a false reading.

CONNECTING IN THE BRIDGE

Two methods of connecting the dummy in the bridge will be described:


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THREE WIRE CONNECTION

The advantage of this connection is that minimal lead resistance (span) error isintroduced. It is useful for single measurement points.

This connection method is widely used for other resistance-type transducers, such asresistance thermometer devices (RTD’s), for example, the very precise platinum wirethermometer (Pt 100).


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FOUR WIRE CONNECTION

Although this connection method introduces double the span error of the previouscircuit, it has the advantage that one dummy (R2 circuit) can be used for a number ofmeasurement points, each of which uses only one active strain gauge (R1).


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METHODS OF COMBINING READINGS FROM MULTIPLE BRIDGES

The reason for combining readings is usually if the need arises to measure the total oraverage strain in a number of members, without regard to the values of the individualstrains.

BEAM A BEAM B

For this example, we need to measure the average (or total) strain on the twomembers shown.

We can, of course use two strain amplifiers to individually measure the strains, thenmix their outputs. However, if the individual strains are not important, we can use amore elegant and more economical method, that is to connect the bridges together inone of the following two circuits, either parallel, or series, as follows:


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PARALLEL CONNECTION OF THE BRIDGES

BEAM A BEAM B

This connection method will reduce the bridge resistance to half of that of a singlebridge. It may mean that the supply current capability of the amplifier will be exceeded.Some strain amplifiers (especially the cheaper ones) can not accommodate a bridgeresistance below 350 Ω without special modifications.

This is the circuit used for connecting multiple loadcells to a single amplifier, such asin platform scales and weighbridges.

The equivalent circuit is as follows:


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BEAM A BEAM B

It can be seen from the equivalentcircuit, that it is essential for theresistances of the two bridges to be thesame if the signal is to be a faithfulindication of the average strain.

SERIES CONNECTION:

Although, at first glance, this connectionmethod appears much more complicatedthan the previous, parallel connection, closeinspection will reveal that it is, in fact assimple. The strain gauges of BEAM A arewired in a half bridge, as are the straingauges of BEAM B. Wiring errors areeliminated if the wiring is kept symmetrical,and the added advantages of the seriescircuit are:

- Bridge resistance is the sum of theindividual bridges, making it easy on themeasuring amplifier's excitation circuit,reducing excitation current

- double the excitation voltage can be used toincrease the mv signal level (especiallyuseful in electrically noisy environments)

This connection is not used for loadcells, because each loadcell is a complete bridge,and the bridge has to be broken to allow connection in series.


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The equivalent circuit for the series connection is shown here:


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FORMULAE

1. QUARTER BRIDGE

UOUT k . ε------ = -----UEXC 4000

Output (mv/v) is equal to Gauge Factor times strain (µm/m) divided by four thousand(the thousand is to convert the micro in the strain to milli in the volt).

NUMERICAL EXAMPLE:

With a gauge factor of 2, and a strain of 500 µm/m, output wll be:

2 . 500 -------------- = 0,25 mv/v 4 000

(See Page 6)

2. HALF BRIDGE

UOUT k . (ε1 - ε2)------ = ---------------UEXC 4000

Output (mv/v) is equal to Gauge Factor times the difference of the strains (µm/m)divided by four thousand.

NUMERICAL EXAMPLE:

For measuring normal strain, with a gauge factor of 2, and strain1 of 500 µm/m(tension), strain2 of -150 µm/m (compression), the output wll be:

2 . (500 – (-150)) -------------------------- = 0,325 mv/v 4 000

(See Page 8)


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3. FULL BRIDGEUOUT k . (ε1 - ε2 + ε4 – ε3)------ = ---------------------------UEXC 4000

Output (mv/v) is equal to Gauge Factor times the difference of the strains (µm/m)divided by four thousand.

NUMERICAL EXAMPLE 1:

For measuring normal strain, with a gauge factor of 2, and strain1 of 500 µm/m(tension), strain2 of -150 µm/m (compression), the output wll be:

2 . (500 – (-150) + 500 - (-150)) ----------------------------------------- = 0,65 mv/v 4 000

(see Page 19)

NUMERICAL EXAMPLE 2:

For measuring bending, shear, or torsion withε1 = +500 µm/m (tension)ε2 = -500 µm/m (compression)ε3 = +500 µm/m (tension)ε4 = -500 µm/m (compression)

2 . (500 – (-500) + 500 - (-500)) ----------------------------------------- = 1 mv/v 4 000

(see Page 10)

This concludes the lecture on strain gauge circuitry. As an attachment, an extract isshown from Mr Karl Hoffman's book: Applying the Wheatstone Bridge Circuit, HBMPublication vd72001e, (reproduced here without permission) which summarises thevarious strain gauge connections for your reference. Other material for this lecturewas obtained from Grolier Multimedia Encyclopaedia (historical notes) and from hardexperience, with MicroMeasurements, HBM KALIBER, and our own, South Africanmade IEM strain gauges, with hundreds of field applications Locally, as well as inAustralia, Europe and the Middle East.

July 1996Modified March 2000Louis Eder, Instruments for Engineering Measurement, Germiston.


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INDEX

altered apparent strain, 24ambient temperature, 17AMPLIFIER INPUT RESISTANCE, 21average strain, 28bending, 12bridge resistance, 29bridges, 22cancellation of unwanted strains, 13cantilever beam, 11COMBINING READINGS, 28compensation, 17compensators, 22compressive strain, 8dummy, connecting, 25equivalent circuit, 23FOUR WIRE CONNECTION, 27full bridge, 11FULL BRIDGE formula, 33galvanometer, 21gauge factor, 4half bridge, 9HALF BRIDGE formula, 32HBM, 33input resistance, 23

linearity, 9non-linearity, 8offset value, 19ohmmeters, 4PARALLEL CONNECTION, 29Poisson, 20Poisson's ratio, 9Pt 100, 26quarter bridge, 6QUARTER BRIDGE formula, 32Resistivity of copper, 14RTD, 26self-temperature-compensation, 24SERIES CONNECTION, 30shear, 12single measurement, 26symmetrical strain, 11Temperature, 13, 16temperature compensation, 20THREE WIRE CONNECTION, 26torsion, 12voltage ratio, 10Wheatstone bridge., 6wire resistance, 14

Wheatstone bridge

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