Seco 4.4 Straln-Senslng Elements 275

4.4 STRAIN-SENSING ELEMENTS

When a mechanical structure is subject to a force, it undergoes deformation.Stress describes the intensity of the force on the structure as the force per unitarea (F/A) and strain describes the deformation as the fractional change inlength (t.LlL). Youngs modulus describcs the stiffriess of lhe structure as theratio of stress to strain. If the structure is very stiff, a large stress is required toproduce a given strain and the Young's modulus is large.

One of the simplest resistive strain gauges is a rubber tube filled withmercury. As the rubber tube is stretched, its length L increases and its cross-sectional area A decreases. The electrical resistance R is given by R = pUA,where p is the resistivity in ohm em, and for constant volume V = AL, R =pU/V. For small changes in length M, and a constant electrical resistivity p, 6.R= 2pL M/V and MIR = 2 M/L is the measure of the strain. The mercury tubestrain gauge has been used for respiration monitoring.

4.4.1 lhe Bonded Resistance Strain Gauge

The metal-foiI strain gauge consists of a pattem of metal on a mylar backing(Figure 4.26) that changes its resistance as it is placed under tension or com-pression (Figure 4.27). This sensor has a very low cost and can be cemented tostructural members of buildings, bridges, boilers, hulls of submarines and ships,etc. They can be used to detect strain in real-time and wam against excessivestrain that could cause material failure. A typical inaximum safe strain is L\LIL =0.5%.

~ Bonding Pad / Long Conductor Path

"'""'-1.---~ Direction of strain sensitivity

Figure 4.26 Metal-foil strain gauge, usually a metal pattem on Mylarfilm. For most metais, the fractional change in electrical resistance (M/R)is approximately twice the fractional change in length (!:::.UL).

276 Sensors and Actuators Chap. 4

U nstrained U nder tension

Length increasedArea decreasedResistance increased

Under cornpression

Length decreascdArea increasedResistance decreased

Figure 4.27 The metal-foil elcmcnt in the strain gauge changcs its shapcduring tension and expansion, resulting in a change in electricai resistance.

When rnetals are placed under tension, their length L will increase but thegirth D will not necessarily decrease to keep the volume a constant, as it wasfor the mercury tube strain gauge. Defining the longitudinal strain ti = Sl.ll.and the transverse strain Et = 'D/D, Poisson's Ratio u is defined as-Et/EI. Because the cross-sectional area A varies as the square of the girth D,M/A = 2 Iill/D = -2u(MJL).

The electrical resistance of a rectangular bar of metal of volume resistivityp is given by

R= LpA

Taking partial derivatives,

dR = E.aL _ pL aA + L ap = R aL _ R aA + R apA A2 A L A P

In differential form,

M = M _M + ,p = (l +2u)M + ,pR L A P L P

The term M/L is dimensional and the term 'p/p is piezoresistive. For theconstant-volume mercury strain gauge, (J = 0.50 and the gauge factor Gs in theexpression M/R = Gs M/L has the value Gs = 2. For most metals, U = 0.30(increase in volume with strain), but the piezoresistive term increases Gs so thatit is also about 2. The gauge factor ranges from 2 to 4.5 for metais and can be

Sec.4.5 Force and Pressure Transducers 277as high as 150 for semiconductors (Table 4.11). The high gauge factor for serni-conductive strain elernents is because the resistivity p is a strong function ofthe strain.

TABLE 4.11 STRAINGAUGE MATERIALSAND GAUGE FACTORS

Material Os :::(L1RIR)1(LlliL)

(61UR)/t\T (LUf,)/6.T

(10-5 1C)

Ni4SCUS5 alloySilicon (n-typc)Silicon (p-type)

2.1100 to 170

-100 to -140

270 to 70070 to 700

1.6

0.230.23

For p-type silicon (p = 2 X 10-2 n-cm), t:..R1R= 120 WL + 4,000 (WL)2

For n-type silicon (p = 3 x 1Q-4 n-crn), /'ill/R = -110 M.JL + 10,000 (M.) L)2

4.5 FORCE AND PRESSURETRANSDUCERS

4.5.1 Force Transducers-

A cornrnon rnethod of rneasuring the force of gravity F on a rnass m uses aspring and position sensor. F = mg = kx, whereg is the gravitational accelera-tion, k is the spring constant, and x is the difference between the loaded andunloaded lengths of the spring. Since the gravitational force does not dependon displacernent, we can use a spring that perrnits a significant displacernent,which helps the accuracy of the rneasurernent. In rnany cases, however, adisplacernent in the direction of the force (i.e. "yielding" to the force) signifi-cantly reduces the magnitude of the force, so it is irnportant to rneasure theforce with a srnall displacement. One of the best methods for accomplishing thisuses the piezoelectric crystal, which produces a voltage that is proportional tothe force. Another method uses one or more strain gauges cemented to a flexi-ble rod (Figure 4.28). One end of the rod is fixed and the other is attached tothe force. The stiffness of the rod is chosen to provide an accurate measure-rnent over the loads of interest without excessive displacernent.

278 Sensors and Actuators Chap. 4Tensile strain (L'lUL > O)

\11ElasticElement

Compressive strain (L'lUL < O)

Fixed end

/ Load

Figure 4.28 Force transducer consisting of four strain gaugesmounted in opposing pairs to an elastic element.

The resistance of the strain gauge is measured by placing it in one arm ofan initially balanced Wheatstone bridge. If a single strain element is used (left-hand side of Figure 4.29), the output will be sensitive to therrnal expansion ofthe elastic element. A better design uses four strain elements in opposing pairs(right-hand side of Figure 4.29). This results in a fourfold improvement insensitivity and a relative insensitivity to temperature changes.

For the single-element bridge (left-hand side of Figure 4.29) with an exci-tation voltage Vb, the output is given by

v - V = V (~_ R ) = V ( M ) ~ V (M) = V (Gs M)+ - b 2R ,2R+M b 4R+2M b 4R b 4 L

For the four-element bridge (right-hand side of Figure 4.29), assumingthat all four strain elements have the same unstrained resistance R, we have

V -V =v ( R+M _ R-M .)=v: (2M)=v (c M)+ - b R+M+R-M R+M+R-M b 2R b S L

Note that the relationship for the four-element bridge is naturally linear,whereas that of the single-strain-elernent bridge is not. In addition, the fourelement force transducer has four times the sensitivity. A typical maximumstrain l.Jl, would be 0.5%, so that for a gauge factor Gs = 2, tlRIR would beonly 1% and Vowould be 1% of Vb.

."

Seco 4.5 Force and Pressure Transducers 279

Figure 4.29 Forcc-transducer bridge circuits usmg a singlc str ainelement RS. ar four strain elernents, 11. 12. b 1. and b2. in opposing pairs.Thc four-elernent bridge has four times lhe sensitivity of the single-element bridge.

It should also be noted that a uniform change in temperature will cause achange in the length of the bar and put an equal strain on all the gauges.Therefore, the single-element force. transducer is sensitive to temperaturechanges, while the four-element force transducer is not.

4.5.2 Pressure Transducers

The pressure of a gas or a fluid is the force per unit area exerted perpendicu-larly on the surface of the surrounding container. One of the most sensitivepressure transducers is the piezoresistive diaphragrn, which consists of fournearly identical semiconductor piezoresistors buried in the surface of a thincircular silicon diaphragm. Pressure causes the diaphragm to bend, inducing astress on the diaphrgm and the buried piezoresistors. Two of the resistorsincrease in value' and two decrease, depending 00 their orientation withrespect to the crystalline direction of the silicon material. Gold pads attached tothe silicon diaphragm provide connection from the piezoresistors to a fullbridge similar to the force transducer described in the previous section. Siliconis elastic throughout its operating range and fails by rupturing. Units are avail-able for measuring pressures from 1 to 15,000 psi and have typical accuraciesfrom 0.1% to 1% of full scale. For measuring absolute pressure, one side of thediaphragm is evacuated and sealed. For measuring differential pressure, bothsides are used.

280 Sensorsand Actuators Chap. 4

4.5.3 Piezoelectric Transducers

The most commonly used piezoelectric materiais are bariurn titanatc anel singlccrystal quartzo

Sensor ApplicationsForce, Accelcration, Prcssure

Ultrasound microphone

Phonograph stylus (sodium potassium tartrate)Actuator ApplicationsMicromani pu lator

Ultrasound ernitterCrystal oscillator (single crystal quartz)Sensor-Actuator ApplicationsAcousiic rangingMedical imaging

Figure 4.30 shows a piezoelectric crystal and its equivalent circuit. Thecharge separation produced in the crystal is given by

q=SqF=SqPA

where Sq is the charge sensitivity.

F=PA

d

A

Figure 4.30 Piczoelectric crystal and equivalem circuit,

Seco 4.5 Force and Pressure Transducers 281

The voltage V produced by this charge is given by:

V = q/C where C = kA/d

and A is the area, d is the thickness, and k is the dielectric constant.Cambining these, we have

V= qd = SqPd =S PdkA k v

For sensing acceleration, an inertial mass is used to convert accelerationinto a force. Typical sensitivitics are !O pC per g or 5 m V pcr g. Larger unitshave ti frequency response frorn IHz to 10kHz (Figure 4.31). Smallcr units canrespond to 1 MHz.

roc::C!)

Vl..

282 Sensorsond Actuotors Chop. 4

For A = 1 cm-, d = 1 mm, we have V = SvPd, which evaluates to:V = (1.2 x I()-4 VrnIN)(1Q---3m)(10.1 x 1()-4 Nm-2) = 12.1 mV

Figure 4.32 shows a charge amplifier commonly used to produce a stepoutput Vo proportional to an input charge q. Vo = q (CICf). The resistor R has alarge value and is used to restore the output to zero over long time periods.

R

Figure 4.32 Piezoelectric crystal integrating amplifier readout

Inertial Navigation

The purpose of the inertial navigaton system is to accurately measure accelera-tions in three directions and integrate them to determine velocity and position.Drift and noise produces errors that increase with time. It is necessary to recali-brate the system at a known location to remove these errors. This approach hasbeen largely replaced by the Global Positioning System (GPS), which uses thetransit time of radio signals to artifical earth satellites in known orbits to deter-mine position.

4.5.4 Vacuum Sensorsand Pumps

The vacuum has many practical applications, inc1uding the following:Cathode ray tube - TV and computer displayPower amplifiers - radio, TV, and radar transmittersIon implantation - integrated circuit productionElectron, proton, and ion accelerators - medical therapy beamsProton and deuteron cyc1otrons - medical isotope production

Seco 4.5 Force and Pressure Transducers 283

Th following are techniques for measuring vacua (see TabIe 4.12 forsurnrnary):

Tabla 4.12 PRESSURE SENSORS AND THEIR PRESSURE RANGES

Pressure sensar . Pressure range

Silicon disk with strain gauges 1000 atm to 1 torr*Capacitancc 1000 to 1 torrThcrmocouplc 1 to 0.01 torrIon gaugej 10-2 to 10-4 torrVac-ion pump 10-4 to 10-12 torr

* 1 atm = prcssure of a 760 mm colurnn of Hg = 760 torrt Bayart-Alpert gaugc can record 10-6 torr

Capacitancc: Paral lel plate capacitor in a rcsonant circuito RcsonanccIrcqucncy em be mcasured vcry accurately- depenes 011 dielcctric constam(proportional to gas pressure) between plates.

Thermocouplc: Heater element is placed below the sensing junction of athermocouple. Convcctive heat transfer to the therrnocouplc - gas prcssurc.

Ion Gauge: A heated filament generates electrons that are accelerated toa (+) collection pIate. These can collide with gas atoms in the way to produce(+) ions that are collected on a (-) collection plate. That current is proportionalto gas pressure.

Vac-Ion Pump: Electrons are generated and spiral in a combined eleetricand magnetie field. These ean eollide with gas atoms to make (-) ions that aredriven into a (+) pIate. That current is proportionaI to gas pressure.

The following are means for producing vacua:

Aspirator: Uses the entrapment of air by a rapidly flowing fluid, such aswater. Limited to 10 mm Hg (10 torr) and is used bychernists in filtering solu-tions. The solution is plaeed above the filter paper, n eontaet with atmosphericpressure, and the aspirator provdes a vacuum in the beaker below the flter.

Bellows and valve: Limited to 1mm Hg (1 torr).Roughing Pump: Mechanical pump (pistons and valves)- can achieve

0.01 torro Lubrication oil a potential contamination- an oil trap is usual1y placedat pump inIet.

Turbomolecular Pump: High speed turbine blades collide with gasmolecules and drive them into roughing pump- operates below 0.1 torr and canachieve 10-s torro

.,'

284 Sensorsand Actuators Chap,4

Vac- Ion Pump: EIectrons are generated and spiral in a combined electricand magnetic fieId. These can collide with gas atoms to make (-) ions that aredriven into a (+) plate. PIate is "degassed" as needed with heat and pumping(usually a turbomolecular pump plus a roughing purnp). Operates below 10-4tOIT and can achieve 10-12 tOIT, '

Diffusion Pump: (Not used in modem systerns.) Boiling oil or Hg issprayed to entrain gas molecules and drive them to the roughing purnp.Operates below 0,1 tOIT and can achieve 10-6 torr. Hot oil can contaminatesystem and limits achievable pressure.

Cryopump: Conventional pumping techniques (such as the vac-ionpump) are limited by the vapor pressure of residual gasses. These residual gasescan be frozen onto a cryogenically cooled surface to produce pressures below10-,12 tOIT.

Note: The 1990 Guinness book of Records states that the best vacuumever achieved was 10-14 torr (using 4K cryopumping and an outer guardvacuum): at the IBM Watson Research Center.

4.6 MEASURING L1GHT

4.6.1 The Silicon Photodiode

The best comrncrcially available photcdiode for lhe measurerncnt of low lightlevels is the PIN photodiode, which is manufactured by diffusing donor (n-type) impurities (usuaIly boron) and acceptor (p-type) impurities (usuallyphosphorous ar arsenic) into opposite sides of a high-purity silicon crystal(Figure 4.33). The crystal is typically 50 to 500 um thick and the P layertransmits light in the wavelength range fram 400 nm to 800 nm. The relation-ship between photon wavelength and energy E is given by:

E = hct); where hc = 1240 eV nm

Photons of energy 2 eV have = 620 nm and are in the red. Photons ofenergy 3 eV have = 413 nm and are in the blue.