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by: Manuela Boyer PhD


Manuela Boyer PhD

GPA 3.66

Russell Tatro

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Russell Tatro
Class Notes
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This 36 page Class Notes was uploaded by Manuela Boyer PhD on Monday October 5, 2015. The Class Notes belongs to EEE 120 at California State University - Sacramento taught by Russell Tatro in Fall. Since its upload, it has received 16 views. For similar materials see /class/218823/eee-120-california-state-university-sacramento in Electrical Engineering at California State University - Sacramento.

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Date Created: 10/05/15
How strain gages work IStrain Stress and Poisson39s Ratio When a material receives a tensile forceP it has a stress 5 that corresponds to the applied force In proportion to the stress the crosssection contracts and the length elongates by AL from the length L the material had before receiving the tensile force see upper illustration in Fig 1 Fig1 g u Tension The ratio of the elongation to the original length is called a tensile strain and is expressed as follows 9 AL 39 9 Strain L Original length AL Elongation See the lower illustration in Fig 1 If the material receives a compressive force it bears a compressive strain expressed as follows AL L For example if a tensile force makes a 100mm long material elongate by 001mm the strain initiated in the material is as follow 9 Thus strain is an absolute number and is expressed with a numeric value plus x10 6 strain us or umm The relation between stress and the strain initiated in a material by an applied force is expressed as follows based on Hooke39s law 6 E9 5 Stress E Elastic modulus 9 Strain Stress is thus obtained by multiplying strain by the elastic modulus When a material receives a tensile force it elongates in the axial direction while contracting in the transverse direction Elongation in the axial direction is called longitudinal strain and contraction in the transverse direction transverse strain The absolute value of the ratio between the longitudinal strain and transverse strain is called Poisson39s ratio which is expressed as follows 82 V v Poisson39s ratio 91 Longitudinal strain or Fig 1 92 Transverse strain or Fig 1 Poisson39s ratio differs depending on the material For refer ence major industrial materials have the following mechanical properties including Poisson39s ratio 0 Mechanical Properties of Industrial Materials G Material IPrinciple of Strain Gages Each metal has its specific resistance An external tensile force compressive force increases decreases the resistance by elongating contracting it Suppose the original resistance is R and a straininitiated change in resistance is AR Then the following relation is concluded AR AL P KS L where Ks is a gage factor the coefficient expressing strain gage sensitivity Generalpurpose strain gages use copper nickel or nickelchrome alloy for the resistive element and the gage factor provided by these alloys is approximately 2 Kse ITypes of Strain Gages Types of strain gages include foil strain gage wire strain gage and semiconductor strain gage IStructure of Foil Strain Gage The foil strain gage has metal foil photoetched in a grid pattern on the electric insulator of the thin resin and gage leads attached as shown in Fig 2 below Example of KFG gage Fig 2 Base Metal foll Lamlnate fllm Bonded surface Gage lead sllveraclad copper wlre 012 to 01mm and 25mm long The strain gage is bonded to the measuring object with a dedicated adhesive Strain occurring on the measuring site is transferred to the strain sensing element via the gage base For accurate measurement the strain gage and adhesive should match the measuring material and operating conditions including temperature For the method of bonding the strain gage to metal refer to Page 8 IPrinciple of Strain Measurement Straininitiated resistance change is extremely small Thus for strain measurement a Wheatstone bridge is formed to convert the resistance change to a voltage change Suppose in Fig 3 resistances Q are Ri R2 R3 and R4 and the bridge voltage V 39 en the output voltage eo V is obtained with the following equation e0 R1 R2R4 E R1 R2 R3 R4 Suppose the resistance R1 is a strain gage and it changes by AR due to strain Then the output voltage is e RiARR3 R2R4 E RiARR2R3R4 fRiR2R3R4R 2 2 e0 R RAR R E 2R AR 2R Since R may be regarded extremely larger than AR 7 4R L e074 R E 4 KseE Thus obtained is an output voltage that is proportional to a change in resistance Le a change in strain This microscopic output voltage is amplified for analog recording or digitial indication of the strain Fig 3 IStraingage Wiring Systems A straingage Wheatstone bridge is configured with 1 2 or 4 gages according to the measuring purpose The typical wiring systems are shown in Figs 4 5 and 6 For varied straingage bridge formation systems refer to Bridgepdf O lgage system With the 1gage system a strain gage is connected to a side of the bridge and a fixed resistor is inserted into each of the other 3 sides This system can easily be configured and thus it is widely used for general stressstrain measurement The 1 gage 2wire system shown in ig 41 receives much influence of leads Therefore if large temperature changes are antici system shown in Fi 4 system refer to quotMethod of Compensating Temperature Effect of Leadwirequot Page 5 Fig 41 Fig 42 Strain gage Strain gage Rg R9 O 2gage system With the 2gage system 2 strain gages are connected to the bridge one each to the 2 sides or both to 1 side a fixed resis tor is inserted into each of the other 2 or 3 sides See Figs 51 and 52 below There exist the activedummy method where one strain gage serves as a dummy gage for temperature compensation and the activeactive method where both ga ges serve as active gages The 2gage system is used to elim inate strain components other than the target strain according to the measuring purpose 2 ga es are connected to the bridge in different ways For details refer to quotHow to Form Straingage Bridgesquot Bridgepdf Fig 51 Dummy gage Active gage Active gage l 0 Active gage O 4gage system See Fig 6 The 4gage system has 4 strain gages connected one each to all 4 sides of the bridge This circuit ensures large output of straingage transducers and improves temperature compensation as well as eliminates strain components other than the target strain For details refer to quotHow to Form Strain gage Bridgesquot Bridgepdf Fig6 0 Typical Measurements with Strain Gages IBending Stress Measurement 1 1gage System As illustrated below bond a strain gage on the top surface of a cantilever with a rectangular section If load W is applied to the unfixed end of the cantilever the straingage bonding site has the following surface stress 5 o so E Strain so is obtained through the following equation 6WL 90 Eth where b Width of cantilver h Thickness of cantilever L Distance from the load point to the center of strain gage 7 L Strain gage Bending Stress Measurement with 1gage System 2 2gage System Strain gages bonded symmetrically on the front and rear surfaces of the cantilever as illustrated below output plus and minus signals respectively with an equal absolute value If these 2 gages are connected to adjacent sides of the bridge the output of the bridge corresponding to the bending strain is doubled and the surface stress 5 at the straingage bonding site is obtained through the following equation 80 o 2 E The 2 gage system discards straingage output corresponding to the force applied in the axial direction ofthe cantilever L Strain gage Strain gage Bending Stress Measurement with 2 gage System IEquation to Obtain Strain on Beams Strain so on beams is obtained through the following equation M 80 ZE where M Bending moment refer to Table 1 Z Section modulus refer to Table 2 E Young39s modulus refer to quotMechanical Properties of Industrial Materialsquot page 6 Typical shapes of beams and their bending moments M and section moduli Z are shown in Tables 1 and 2 Table 1 Typical Equations to Obtain Bending Moment Shape of Beam Bendinq Moment M W i4 MLL OSL52 M2 4 e 2 we Li OL i aMiiw zer 3 TsLseari 74g W 0 M T L 7M L2 Me 4 L 5Ls AM7Wl L L L z 2 w 2 1w H m OSLSli ZiSLSwi 224gtMW21 Table 2 Typical Equations to Obtain Section Modulus Cross Section Section Modulus Z 1 2 6bh bh23 hia h2 ml e I Torsional and Shearing Stress Measurement of Axis When twisted an axis has shearing stress 17 and in the 2 directions inclined by 45 from the axial line it has tensile and compressive stress in an equal magnitude to the shearing stress In measuring strain on a twisted axis under simple shearing stress status the strain gage does not directly measure the shearing strain but detects tensile or compressive strain initiated by tensile or compressive stress that is simultaneously generated with the shearing stress Stress conditions on a microscopic part of the surface of the axis may be as illustrated below Tensile stress 0 Shearing stress 1T Compressrve stress 0 Ir a r Shearing stress 7 is defined as illustrated below and the magnitude is calculated through the following equation L 7 G where G Shearing modulus refer to quotMechanical Properties of Industrial Materialsquot page 1 TI Shearing stress When the axis is twisted point A moves to point B thereby initiating torsional angle 8 g 27 22 d d 2 1 Stress Measurement with 1gage System Bond the strain gage on the twisted axis in the direction inclined by 45 from the axial line The relation between strain 90 and stress 5 is expressed with the following equation to obtain tensile or compressive stress 5 goE 1v 5 where so Indicated strain E Yo g39s modulus refer to quotMechanical Properties of Industrial Materialsquot page 1 v Poisson39s ra io Stress c and shearing stress 17 are equal in magnitude and thus 17 s 2 Stress Measurement with 2 or 4gage System 2 or 4 strain gages forming the straingage bridge are strained in an equal magnitude to enable 2 or 4 times larger output Accordingly the stress is calculated by dividing the indicated strain 4 For axial strain measurement the 2 or 4gage system is used to eliminate strain caused by bending moment Also for meas urement of tensile strain and compressive strain strain gages are symmetrically positioned from the center of the axis as shown below Tensile strarn Compressrve strarn 3 Application to Torque Measurement Strain on the surface of the axis is proportional to the torque applied to the axis Thus the torque is obtained by detecting the strain on the surface Shearing stress distributed on the lateral section is balanced with the applied torque T establishing the following equation where Zp Polar modulus of section This equation may be rewritten as follows by substituting the shearing stress with the relational expression of tensile strain and stress T so E Zp 1 v The polar modulus of the section is specific to each shape of the crosssection as follows Cross Section Polar Modulus of Section Zp tie aw A straingage torque transducer can be designed using the aforementioned relational expression of so and T Obtain so from the allowable stress for the material and determine the width d of the axis which is matched with the magnitude of the applied torque Then amplify the strain out put with a strain amplifier and read the output voltage with a measuring instrument Principle of Selftemperaturecompensation Gages SELCOM Gages Suppose the measuring object and the resistive element of the strain gage have linear expansion coefficients is an the object provides a thermallyinduced apparent strain 8T per 1 C that is expressed with the following equation ew HES g here ot Resistive temperature coefficient of resistive element Ks Gage factor of strain gage Resistive element g Resistive element 135 The selftemperaturecompensation gage is designed so that ST in the above equation is approximated to zero by controlling the resistive temperature coefficient of the age39s resistive element according to the linear expansion coefficient of the measuring 0 39ec hen bonded to a suitable material KYOWA39s selftempera turecompensation gage M gage minimizes apparent strain in the compensated temperature range to 18 e C graph below shows apparent strain output of 3wire strain gage we co Temperature v so 100 I I L I co a M lt2 lt2 lt2 lt2 lt2 Thermallyainduced apparent strain output uwc Typical Temperature Characteristics of Selftemperaturecompensation Foil Gage 0 Linear Expansion Coefficients of Materials 0 Temperature Effect of Leadwire with 2 wire System 39 quot quot 39 39 39 39 prdlelll ouam Laimre Area of Conductor of 1 0m long Leadwire with 10m Extension mm approx 9 x ud C L5 05 07 1 1 3 L9 01 1 32 506 L6 008 44 690 1209 gage Thermallyinduced apparent strain Sr us C is obtained through the following equation 9 U L Rg r Ks where Rg Resistance of strain gage 9 r1 Resistance of leadwire Q Ks Preset gage factor of strain amplifier usually 200 ct Resistive temperature coefficient of copper WIre ARR C 39 x10 3 Method of Compensating Temperature Effect of Leadwire 3wire System For effective selftemperaturecompensation SELCOM gages adopt the 1gage system However if the leadwire cable is a 2 wire system strain output from the bridge is affected by temperature effect of the leadwire To avoid such adverse effect the 3wire system is adopted lf 3 leads are connected to the strain gage as shown below one half the leadwire resistance is applied to the adjacent side of the bridge to compensate the resistive components of the 2 leads affected by a similar temperature change and thus the bridge output is free from any temperature effect of the leadwire The temperature effect of a third lead connected directly to the amplifier can be ignored since the amplifier provides a high input impedance As precautions in using the 3wire system the 3 leads should be the same in type length and crosssection to receive the same temperature effect If they are exposed to direct sunlight the coating color too should be identical R3 R2 R9 7 rt 1 RI 7 r1 v 7 rl 0 In uence of Insulation Resistance The insulation resistance of a strain gage including leads does not affect the measured value if it is higher than 100MQ But if the insulation resistance changes drastically during measure ment it causes the measured value to include an error n gtr2 Insulation resistance Input Bridge Circuit Designed with Insulation Resistance Taken into Consideration If the insulation resistance descends from I 1 to r2 in the figure above error strain 9 is g Rg I 1 r2 Ksl 1l 2 Suppose Rg 1209 resistance of strain gage Ks 200 gage factor of strain gage I1 1000MQ original insulation resistance r2 10MQ changed insulation resistance Then the error strain is approximately Gus In general strain measurement such an error causes virtually no problem In practice however the lowered insulation resist ance r2 is not kept constant but sharply changes due to temperature humidity and other conditions Thus it is not possible to specify to which part of the circuit the insulation resistance r is applied Accordingly precautions should be taken 0 Resistance Change of Strain Gage Bonded to Curved Surface The strain 8c occurring on the resistive element of a strain gage bonded to a curved surface may be expressed with the following equation t SC 2rt where t Thickness of gage base plus adhesive layer r Radius of gage bonding surface For example if a uniaxial KFG gage of which the gage base including the adhesive layer is 0015mm thick is bonded to a curved surface of 15r e strain gage receives strain of approximately 5000ug under the mere bonding condition If the gage factor Ks is 200 ARR 3910000ug since ARR g Ks If the gage resistance is 1209 it increases by approximately 129 If the gage is bonded inside the curve the resistance decreases Resistive element A Gag e base 11111111111111 IIIIIIIIIIIIIIIIIIII I III II quotlungquot Strain Gage Bonded on Curved Surface Method of Compensating Gage Factors If the gage factor of the strain gage is different from that 200 of the strain amplifier the real strain 9 can be obtained through the following equation 8 X 8i where 9 Measured strain Ks Gage factor of strain gage Misalignment Effect The strain 90 measured by a strain gage that is misaligned by an angle Sfrom the direction of the principal strain is expressed with the following equation 90 91 92 81 92 cos 28 If 92 V 1 v Poisson39s ratio under the uniaxial stress condi tion 90 l2 911 v 1 v cos 29 Method of Compensating Leadwire Extension Effect If the leadwire or cable is extended with the 1gage or 2 gage system additional resistance is initiated in series to the strain gage thereby decreasing the apparent gage factor For example if a 10m long leadwire with 03mm2 conductors is used the gage factor decreases by 1 In the case of the 4 gage system transducer the extension decreases the bridge voltage too In these cases the real strain 9 can be obtained through the following equation Supposing the gage factor Ks is 2 00 39 g1Rxgi where 9 Measured strain Rg Resistance of strain gage r1 Total resistance of leadwire For reciprocating resistance see the table on the next page Oneway resistance in the case of 3wire system Reciprocating Resistance of Leadwire CrossSection NumberDiameter BeCiPmca ing mmZ Strands Resistance per 10m Remarks 9 008 7012 44 L6 7 011 10012 32 L9 10 03 12018 117 L2 05 20018 07 L5 0 Method of Compensating Nonlinean39ty of 1gage System Nonlinearity beyond the specification in large strain measure ment with the 1gage system can be compensated through the following equation to obtain the real strain 9 90 9 1 90 x10 where so Measured strain 0 Method of Obtaining Magnitude and Direction of Principal Stress Rosette Analysis Usually if the direction of the principal stress is unknown in stress measurement of structures a triaxial rosette gage is used and multiple physical quantities are obtained by putting measured strain values in the following equations These equations apply to rightangled triaxial rosette gages Precautions in Analysis 1 Regard ea gt b gt Sc as the forward direction 2 Angle 0 is Angle of the maximum strain to the ea axis when ea gt 9c Angle of the minimum strain to the ea axis when ea lt Sc Comparison between 9a and Sc in magnitude includes plus and minus signs ea 8c 28a 8b2 8b ec2 ea 8c 28a 8b2 8b ec2 Max principal strain gmax l 39 2 Min principal strain mm Direction of principal 6 1 tanil 28b ea 8c ea 8c strain from 8a aXIs 2 Max shearing strain ymax 2sa 8b2 8b Sc2 Max principal stress Gmax 21 EV 1 vga 8c 1 V x 2isa sb2sb sc2 Min Principal stress 5min wig n vsa Sc 1 v x 28a 8b28b Sc2 Max shearing stress Tmax 72 E V x 28a 8b2 8b ec2 v Poisson39s ratio E Young39s modulus Refer to quotMechanical Properties of Industrial Materialsquot page 6 Generating Calibration Value based on Tip Parallel Resistance Method When extending the leadwire by several hundred meters or to obtain an accurate calibration value use the tip parallel resist ance method The parallel resistance r can be obtained through the following equation r ii Ks e where Rg Resistance of strain gage Ks Gage factor of strain gage 9 Calibration strain value Examples of Calibration Strain Value and Resistance Rg 120Q Ks 2 00 Calibration Strain Value Resistance r approx 100 us 600 kg 200 us 300 kg 500 us 120 kg 1000 us 60 kg 2000 us 30 k9 Typical Strain Gage Bonding Method and Dampproofing Treatment The strain gage bonding method differs depending on the type of adhesive applied The description below applies to a case where the leadwireequipped KFG gage is bonded to a mild steel test piece with a representative cyanoacrylate adhesive CC33A The dampproofing treatment is in the case of using an butyl rubber coating agent AK22 quotStrain Gage Bonding Manualquot is available from KYOWA at a price of 1 200 per copy Like drawing a circle with sandpaper 300 or so polish the strain gage bonding site in a considerably wider area than the strain gage size Then polish with sandpaper Using an absorbent cotton gauze or SILBON paper dipped in a highly volatile solvent such as acetone which dissolves oils and direction to remove oils and fats Reciprocated wiping does not clean the surface After cleaning mark the strain gage bonding position Make sure of the front metal foil part and the back of the strain gage Apply a drop of adhesive to the back and immediately put the strain gage on the bonding site Do not spread the adhesive over the back If so curing is adversely accelerated Cover the strain gage with the accessory polyethylene do not put it up to adjust the position When the adhesive is cured remove the polyethylene sheet and check the bonding condition Ideally the adhesive is slightly forced out from around the strain gage If the adhesive is widely forced out from around the gage base remove the protruding adhesive with a cutter or sandpaper Place gage leads in a slightly slackened condition Put up the leadwire from before the part where the adhesive is applied Place a block of the coating agent below the leadwire with gage leads slightly slackened Completely cover the strain gage protruding adhesive to pieces but slightly flatten it with a finger to closely contact it with the strain gage an part 0 t e leadwire Completely hide protrusions including gage leads behind the coating agent If required contact your KYOWA sales representative Chapter 3 Noise and Coherent Interference in Measurements Seruon 3 2 Random Noise in Circuits In this chapter the noise is assumed to be timeinvariant That is the physical phenomena creating the noise does not change with time For example a current at time t 0 starts passing through a resistor The resistor will dissipate power The resistor temperature will slowly increase Thus the resistor s resistance and thermal noise will also increase over time Seroor 3 2 1 Probability Density Functions The probability demity function PDF considers only the amplitude statistics ofa stationary noise waveform nt The PDF does not deal with how the noise varies over time p06 Probability thatx lt n S x dx ix Where x is a speci c value ofthe noise n taken at some time t And dx is the differential increment in x I pmdr prob x Sv Seruon 3 1 Introduction L 4 J mnauun 39 1i p 39 39 of measurements and the detectability of the quantity under measurement Noise arises in a circuit from completely random phenomena Random noise comes from within the measurement system As a purely random event it will be assumed that the noise has a zero mean It has no dc value nherent 39 such as power line radio television computer and so on sources Coherent noise enters a measurement system from without Seruon 3 2 Random Noise in Circuits The random noise will typically be statistically described Such as the probability density function cross and autocorrelation functions cross and autopower density spectra and root power density spectrum These characterize the amplifier equivalent input noise Saloon 3 2 1 Probability Density Furrruorrs Properties of the PDF note the limits of integration j pmdr prob x Sv Tpxdx prob v ltr sz T pmdr prob x g no Section 3 2 1 Probability Density Functions Section 3 2 1 Probability Density Functions There are several avors of probability density functions Rayleigh PDF Gaussian noise x2 2 X 2 1 x x 3205 x e 20x2 p 0x 5 Rectangular PDF It is common to assume that noise signals are Gaussian in nature i for a lt x lt a 1906 lt 2a KO for x gt a Section 3 2 2 The Power Density Spectrum Section 3 2 2 The Power Density Spectrum The power density spectrum PDS is de ned as the Fourier transform For example a bandwidth limited noise voltage source has the following of the autocorrelation function cumulative mean squared noise characteristic Recall that the autocorrelation function 1 6tu1 ns any periodicity found w in the data stream 1 T auto correlation Rm 239 lim 5 J ntnt 239o t 00 f T The onesided noise power density spectrum is the derivative or slope The twos1de PDS 1s the Four1er transform of the autocorrelatlon fct Ofthe mean Squared noise Characteristic curve 3quot f A PDS C13moi JRnnTeijdT 2 w The autocorrelation is an even function xt xt thus due to symmetry we can examine just the positive frequency f 20 00 7 Section 3 2 2 The Power Density Spectrum Section 3 2 3 Sources of Noise in Signal Conditioning Systems White noise is by de nition present at all frequencies with a constant Noise in signal conditioning system can be separated into two major mean A categories gt I I Noise from passive resistors an MSVHz JSWUMf 00 Noise from active circuit elements v O sz The above graph implies that the noise power is in nite which is not possible for physically realizable Systems Usually the noise may be assumed to be Gaussian and be characterized as white noise over at least a signi cant portion of the frequency Thus 10W frequmcy behavior may be approximated by a 1 f noise Spectrum power density spectrum 1 1 i an 1 EE 239 lf noise can be thought of as a carrier transport issue and is material dependent quotmy a1 E F Elquot ra Section 3 2 3 1 Noise from Resistors A real resistor can be modeled as an ideal resistor in series with a zero mean voltage source R1 Hz V n 8f 4kTF1 R2 MSVHz 0 As the frequency increases past some point the resistor will no longer be a linear lumpedparameter element But for low frequencies the resistor s noise is wellmodeled by a Gaussian white noise source Noise from resistors is called thermal or Johnson noise Section 3 2 3 1 Noise from Resistors In terms of current the PDS is 4kT MS Amps R HZ Snf Under the condition of a dc or average current the Johnson PDS is modi ed as follows A2 MS Volts HZ Snf4kTR Where I is the dc or average current through the resistor The parameter A is a constant that depends on the material from which the resistor is constructed such as carbon wire or metal lm The 1f noise is best handled by keeping the wattage rating of the resistors as high as practical See gure 35 on page 117 Section 3 2 3 2 The Two Source Noise Model for Active Devices The two noise sources e1181 and i model the noise behavior of the circuit as shown below 11219 397 II ena RMS VNHz 7 Ina HMS ANHZ nvNHE fANFE p 20 White region 10 10 k 0 l I I I I 39 I 039 1 10 102 103 104 105 106 107 sz There is a white noise equivalent behavior equal power at all frequencies over some range of bandwidth for the speci c amplifier Section 3 2 3 1 Noise from Resistors The onesided power density spectrum for a resistor is s f 4kTR MS V for OSfSoo MSV mean squared voltage Boltzmann s constant k 138 X 103923 JK Temperature T in kelvin In a given bandwidth the mean squared noise from a resistor is f2 V3B SHOW 4kTRltf2 f1 4kTRB Ms Volts f1 Where B is the equivalent noise bandwidth in Hz Section 3 2 3 2 The Two Source Noise Model for Active Devices JFETs BJ Ts and other IC ampli ers are generally described by the two source input model The short circuited input noise voltage ena models all internal noise sources when the input is short circuited The model also needs a current noise source iIla to account for the circuit behavior RST VS C13 in ViHs oV P E O O Section 3 2 3 3 Noise in JFETs The theoretical ie lower limit thermal noise generated in the conducting channel of a J FET is given by 2 4kT 4kT IDSS MS Volts quot0 gm gm I HZ 0 DQ Where gmo is the small signal transconductance measured when VGS O and ID IDSS IDSS is the dc drain current when the device is in saturation that is When VDS Z VGS Vthreshold39 IDQ is the quiescent dc drain current at the FETs operating point Secaon 3 2 3 3 Noise in J39FETs A more practical model that better predicts the low frequency behavior of a FET is 2 4kT1 g Where fE is the measured comer frequency of the low frequency 1f spectrum MS Volts Hz The exponent n ranges between 1 lt n lt 15 and is found by curve tting for a specific device One possible reduction in the ena noise is to heat sink or actively cool Secaon 3 2 34 Noise in BJTs The values of ena and ina in a BJT amplifier stroneg depend on the device operating Q point a BJT small signal model BJT bias circuit The small signal model on the right models the weak base currents that result from the voltage controlled current source gmvbe MSA 5 13 211151 HZ Shot noises In a BJT MSAmps 1 2 31 t q in HZ Secaon 3 4 Broadband Noise Factor and Noise Figure ofAmplifiers The noise factor F of an amplifier is defined as Maw at the ampllfla input NRgmat the ampli er output A real amplifier is noisy It amplifies both the intended signal and any noise in the system within e bandwidth ofthe amplifier gt SN39R np ut uutput Thus the noise factor F gt 1 The mire gure NF is defined as NF 2 lologmF ere F is the mean squared value and taken to be a power representation Noisy amplifier model Secaon 3 2 3 3 Noise in JFETs The author brie y mentions shot noise in this section The charge carrier ofa signal whether an electron or photon has a discrete energy Thus an individual carrier arrives as a packet of energy When very weak signals are present this discrete carrier arrival is now noticeable and the random nature ofthe arrival is detectable The random over short time intervals arrival thus has a shot or quantization effect on the signal This is shot no39 e Section 3 3 Propagaaon of Gaussian Noise through LinearFilters One reason to characterize noise in terms ofa Gaussian white noise is that the output oflinear filters results in Gaussian white n 39 The noise output ofa linear filter does have frequency effects from the filtering which is described by the gain squared bandwidth product Gainz W hymngt12 df n See table 31 for a short list of equivalent noise bandwidths of five common filters Secaon 3 4 Broadband Noise Factor and Noise Figure ofAmplifiers The source Vs noise and the input resistor R5 thermal noise are modeled above along with ena and ina The author lets the source Vs noise zero The input thermal noise is NW 4kTR5 MS Volts The amplinei 39 39 39 function is NEW 4km are tmR JK B MS Vattx Seeuor 3 4 Broadband Noise Factor and Noise Figure ofAmplifiers Seeuor 3 5 Spot Noise Factor and Figure The noise factor for this simple r 39 r 39 a certain noisy ampli er model is then 7 frequency range Read this section as background mainly to understand data sheets in the NW 4km e imR 2 1 K55 future NM 4kTRJB quot L L 39 ofRscan39 39 1 Without loss ofgenerality we can let the gain Kv 1 thus the equation WhiCh depend 0 frequemY see gure 313A 0 Page 128 simplifies to F 1 e392 2R3 In the data sheets the input resistance Rs bandwidth over which the quoted F was determined and the temperature if different than 300 k must be specified Seeuou 3 6 Transformer Optimizauor ofAmplifierFsM and Output SN39R Seeuou 3 7 Caseaded Noisy Amplifiers For an ac input the noise and SN39R may be improved with the use ofa The author shows in this section that it is the first amplifier headstage low noise and lowloss transformer 39 in the cascade series that has the most impact on the noise that m l 04 Vt Thus the lowest noise amplifier should be used first f e a K At 33 mm I I I i i Hquot 39 7 Practical experience also shows that the first stage should have a gain 7 Vagina 7 7 7 7 7 7 e e 7 e a of at least five as well The transformer is a bandpass device It loses efficiency atboth low and high frequencies Thus the general effect of the transformer to shift the minimum spot noise to a useful frequency which depends on the amplifier circuit 5 Section 3 8 1 Calculation of the Mnimum Resolvable ae InputVoltage to aNoisy1uveruug eeuor 3 8 i OprAmp Ampii er Itis quot 39 quot N 1h d mt measurde I oisy inverting opamp w1 sinusoi input ostpeak at frequency f5 y We may address the matter in terms of signal to noise ratios as well i mm mpg Valium The en ve 560 gwe e p1650fmeyature Ofme else Md us The noisy addition to the opamp is the shortcircuit input white noise effect on the noise limited minimum input signal level ource en2 39 L i not included 39 39 39 feedback makes that noise much less than em The mean square output signal voltage is s e w Watt quot 1i Z 1 Section 3 8 1 Calculation of the Minimum Resolvable ac Input Voltage to a Noisy Inverting OpAmp Ampli er n T H in 1 p The total mean squared output noise I quot3932 x 621 m 18 ER W a A l r 11 R 2 R 2 N0 4kTR1 B 4kTRFB e I FF B MS Volts 1 1 K3 The noise equivalent bandwidth B is skipping the derivation 1 7 B 4 fT HZ T R closed 00p 2 1 F R1 Where fT is the unity gain comer frequency of the opamp Example page 133 RF 2109 W I R1 1 0009 H in I 1 Use the following values and nd the peak I quotIV2 a signal input voltage for SNR 1 61132111 em 10 y m 2 W 11 1 cf 1 r m1quot Signal SNROMW 2 Name 1 f 12MHZ By de nition SNR 1 is the lowest detectable signal 4kT 4138gtlt10 V2 23 i3001lt 1656 gtlt1020J s amp K 2 2 1 2 2i R1 R 2 R 2 F 4kTRFeja1Fj 7Z39fT R1 R 2 1 F 7r R 1 RF 1 R1 SNR 1 4ch1 1 4ch1 RF 1 V2 S 2 j 4kTRF e RF t 11211311 1 l 2 RF R1 Example If the unity gain frequency fT 12 MHz then the ampli er comer frequency with gain is 6 12 106H 6 fczl2gtlt10 Hz x 5 z l2gtlt10 Hz ZIZXIO HZ ZIZOkHZ RF 10 100 R1 103 The author s exact calculation found fC 5 9 kHz Section 3 8 1 Calculation of the Minimum Resolvable ac Input Voltage to a Noisy Inverting OpAmp Ampli er Fi itquot I 139 1 The output mean squared SNR is 2 12121 21121 SNR Signal 2 R1 R1 0 Noise R 2 R 2 4ch1 F 4kTRFega 1 F 7rfT R1 R1 The above equation is valid if the input signal frequency is much less than the comer frequency of the closedloop ampli er Example Now substitute in the values 105 W n 103 2 2 166x1020J1000 166x10 20J10510gtlt1029 7rl2gtlt106Hz t 1111 ll 166x10 20J100010 1 66x10 2 I10510gtlt10 17 g110027z12x106Hz 10 21100 2 21 S 105 3 10 n 103 2 166x10 13 166x10 15 3 846x10 5 2 O2gtltlO6 2 Volts2 3 808x10 Volrs2 VS 3 808x10 Volrs2 6 17x10 6 Volts The lowest detectable signal is about 6 2 11V Section 3 8 2 Calculation of the Minimum Resolvable dc Current in White and lf Noise A very small DC current is to be measured using an electrometer op amp transresistor circuit An electrometer opamp is designed so that the input bias current is less than 103913 A and input resistance is greater than 1013 Q R T e LPF I78 GS T I ina EOA out 1 32602 szgwnH 3911 C8 The dc current source is represented by the Norton equivalent circuit on the left above The thermal noise from GS RF and iIla is assumed to have a white spectrum white gt equal noise power at all frequencies The noise from eIla however has a low frequency dependence a 1 f component Section 3 8 2 Calculation of the Minimum Resolvable dc Current in White and lf Noise The author models the noise voltage source eIla as eltfgt as 3 MS V whmise f HZ w J low f eq eet The power density spectrum PDS of the ampli er s output noise is then S 2 b RF 2 2 2 Sn0fena0 7JEIR 1kTGF GSzmRF The ampli er output noise is broadband Now lter this noise signal with a secondorder lowpass lter 1 RFT S s 1 a n ena LPF VO 1 Voui wz 8 l a G3 T l quot3 EOA 3200 2 32 an1 2 n 2 S S2560 on Section 3 8 2 Calculation of the Minimum Resolvable dc Current in White and lf Noise The mean squared dc output voltage signal is S02ISZR MS Volts The following example will examine this output signal versus the noise function on the previous slide Example RS 2 107 Q ena LPF VO 1 Voui ls l J GS T l quot8 EOA 52002 329 1 S 2 2 N0eja0111 Fj 4kTGFGSiaR blnl MS Volts For the rst term where em10 is in units of Vrms example text in error 2 10 2 e2 1R F 35x10 9V 2 110 1227x10 9V2 l227gtltlO9MSV 11610 R ms 107 ms S Section 3 8 2 Calculation of the Minimum Resolvable dc Current in White and lf Noise In table 31 on page 125 item 3 in the s jo form the equivalent noise bandwidth B is BzwnHz 85 The total mean square output noise voltage is the integration of the PDS function Sn0f from zero to the bandwidth B shown above Mathematically the integration results in in nity due to the b f term This is not physically possible and the cure is to ignore the initial time out to some steady state time which depends on the system The noise is then the following rather complicated function 2 2 N0e a0111 F 4kTGFGSijaR blnl j MS Volts S S Example page 134 RF 210109 R T Use the AD549 electrometer op R 21079 em A LPF amp made by Analog Devices S V 1 vow a 8 l 6 GS T l ina EOA 3200 2 32 an1 W n 2H2 85 Find IS for MS SNR 4 given the following values 4kT 4138 gtltlO23 30010 1656gtlt10 20J V A e 35x10 9 ms 139 02gtlt10 15 ms b136gtlt1017 MS Volts naO JE na JE First nd the output noise for the above values 2 2 N0eja0111 Fj 4kTGFGSiaR blnl1 MS Volts S S Example RF 10109 R T 7 Q W LPF a R ena 2H2 S 8C Vo 1 0gtu1 Isl J GST l39m EOA 52002 82 an1 S 2 2 N0e a0111 F 4kTGFGSiaR blnl1 MS Volts For the second term Note that Joule V A sec 4kTGF GSifa13922 amp F85 2 l656x10 20JlO 1OUlO o02x10 j JE 1 658x10 4x10 32 Azms1020 222Hz HZ 1020922Hz 3 3l6gtltlO 78gtltlO 12 Ao2 3 316x104 MSV Example a n 2H2 8 RF 1010g2 RF T RS 2 107 Q ena LPF VO 1 Voui Si N G3 T I ina EOA 8200 2 32 an1 For the third term 2 blngigl11 Fj 1 36x10 MSVln8lOOl2 283gtlt10 MSV S So the output noise voltage is NO 1227x10 9 3 316x104 2 83gtltlO MSV 3 329x10 7MSV Notice that the thermal Johnson noise in the resistors RS and RF clearly dominate the noise Section 3 8 3 Calculation of the Minimum Resolvable ac Input Signal to Obtain a Speci ed Output SNR in a Transformer Coupled Tuned Ampli er This section will not be covered Section 3 8 4 Calculation of the Smallest ARR in a Wheatstone Bridge to Give a Speci ed SNR0 The output of the bridge is ampli ed by a low noise differential instrumentation ampli er followed by a secondorder bandpass lter The purpose of the lter is to pass the ampli ed ac output signal of interest while limiting the bandwidth of the unwanted noise signal By reducing the noise the output SNR is improved BPF jco2 an v jw2an2 ja2 an 1 Vst vs Simeon 1 Example page 134 RF 1010 9 R T r RS 2 107 Q ena LPF VO 1 Voui Isl 1 636 T I ina EOA 3200 2 32 an1 The condition was to nd IS for a MS SNR 4 Signal 2 MS mo Noise N0 SNR 4 Signal 50 4N0 J43 329x10 7MSV J1 332x10 6MSV 1 154gtltlO3Volts The dc output S0 is given by So 1st 1 154 1031 1S 2i 2 1X0 1154x10 13A R 10 o F The text answer is in error Section 3 8 4 Calculation of the Smallest ARR in a Wheatstone Bridge to Give a Speci ed SNR0 We now calculate the smallest incremental change in resistance AR that can be detected in an acpowered Wheatstone bridge circuit BPF v jw2 wn V jw2an2 jco2 an 1 V3t vs sinan 2 Notice that two arms of the Wheatstone bridge are equal and xed The two other arms of the bridge are designed so that the variable resistance increases one leg and decreases the other leg by equal amounts This models a type of strainstress gauge Section 3 8 4 Calculation of the Smallest ARR in a Wheatstone Bridge to Give a Speci ed SNR0 The differential ampli er s two inputs are Negative terminal input A xed voltage divider with resistors R and R v 2vs Positive terminal input A variable voltage divider set by the strain on the sensor V VA V 5 s RAR V RAR 11 AR R ARRAR S 2R S When AR O the voltage V A 12VS Thus the common mode with perfectly matched resistors should result in no output from the differential ampli er assuming no noise BPF v jwZCwn V0 ja2con2 jco2 an 1 v30 vS sincon t Seem 3 8 4 Calculation of the Smallest ARR in aWireaisiorie Bridge to Give a Speci ed Seem 3 8 4 Calculation of the SmallestARR in a Vlheatstone Bridge to Give a Speci ed SN39RD sm The differential amplifier s output ignoring noise for a moment is en 1 AR 1 Vi KilVrVJ Ki l7V 7V Where Kd is the gain ofthe ampli er So the lter output is given by AR EV V176 Vntgt The mean squared signal output SD is The input signal is given as Vst Vs sin27r400t V Recall that the rms value ofa sinusoid is j The center frequency of the bandpass filter is set at 400 Hz The noise is assumed to be the white thermal noise ofthe bridge resistors and the equivalent input short circuit voltage noise em From table 31 on page 125 the equivalent Hz noise bandwidth noise is My 2 2n Ki ZQ quotHz w v5 110 i Seeuori 3 8 4 Calculation of the Smallest ARR iri aVIheatstone Bridge to Give a Speeified E 5mm Xample page 136 Letthe SN39R10 and ndthe re uired ARR The mean squared output noise is the sum ofthe MS noise sources q Given Vspeak 4 V f 400 Hz For the resistors S f 4kTRW5 MSV For the each branch ofthe bridge the equivalent resistance is R 2 rov this by deactivating the voltage sources and finding the resulting parallel resistance SNRD10 Ni 4k r 4k r efn1ltmfquot MSV ZQ 2 2 4kTReK TheMS SNR is then 10002 E 2 7 2 4 R 7 2 3 2 27400 l xlo Jl00005x10 7V1000 26 Hz V55 4 416er2 Example page 136 Example page 136 Let me SN39R 10 and nd the required AR R AR 2 2864 10 gtlt 4V2 10002 E 2 R 10 AR22864x10 z 2864x10 710000 2 2864X10quotQ m s V 2 2 M400 166X10 J100005x10 E 11000 25 Hz The output voltage omie differential ampli er is m s 2 2 M400 101 66X10 Jl00005x10 7171000 25 4V2 10002 2 4 3 Vn KHEVJ 10007Z zgsxlo 04 VW ZR 200000 2 17V 4 572x10quotV 046 mV 27r40Hz HZ V2 10x10 166gtlt10477Z 5x10 Hz 2x10 V2 However die ow omie example is well demonstrated One ofus is wrong 5416X10 27r40Hz 5 ZZ76X10 quot z 2864x10 7 Seeti n s 5 Determination ofthe Conditions for Maximum Output SN39R Given a ection3 9 Modern Low Noise Ampli ers for Use in Instrumentation Signal Simple Inverting opAmp Ampli er with Known em and in Conditioning Systems See Table 32 on page 138 for a list of low noise IOAs This section will not be covered Section 3 10 Coherent Interference and its NIinimization Section3 10 1 Sources of Cohermt Interference In this section the author considers the effect ofperiodic or coherent There are four major sources ofcoherent interference nois e 1 Power line frequency interference such noise is characteriaed by narrow band power density spectrums 2 Radio frequency electromagnetic inte ference RFD w1th harmonic peaks at integral multiples of a fundamental frequency 3 Switching or generatormotor interference This is usually transient en thi periodic noise signal is in the samebandw1dth as the signal being measured then the measurement is degraded 4 So called crosstalk in the measurement system itself Section 3 10 1 1 Direct Electrostatic Coupling ofCoherent Interference Section3 10 1 2 Direct Magnetic Induction ofCoherent Interference Read this section as a refresher on electromagnetic coupling Read this section as a refresher on electrostatic coupling Seetlun3 ID 1 3 GruundLuups Seeuun3 ID 2 Cures furCuhermzlnterfamee enever a ground eondueuon path ls foreed to earry apovver supply return eurrent whlch has a srgnrfreant hlgh frequency eornponent added to the de ground hen rtrs posslble forthe ae povver eornponent ofthe ground eunentto am a 5m we am 391quot 5 bmf m e mPEdan es f 391 1n thrs seetron vve wlll bnefly examlne afevv sourees and the eure to ground path v rnlnlrnlze the eoherent lnterferenee Thus the ground pornts are not at the sarne potenual Thls ls ealled a groundloop Seeuun 3 ID 2 l FUWa39LlneLuwFassFlltas Seeuun 3 ID 2 2 TmnslentVultage Suppressurs There may be hlgh frequency nolse on the powerllne whlch frlters ean Transrent voltage suppressors are usedto prevent the hlgh voltage attenuate splkerllke transrents oeeurnng on the powermams rnput to an The rntentto ertherbloekthrs hlgh frequency nolse frorn entenng your lnsh39umEnt systern frorn causlng physlcal darnage systern by vvay ofthe povver lrne orto prevent some systern puttrng thrs h gh frequency merfmnce on me powerlme vartehrng of rndueuve loads ean eause voltage splkes 50 volts above the rnstantaneous llne voltage vvrth a durauon ofseveral us Multrstage LC lowrpass frlters ean be usedto attenuate eoherent Whlle not generally harrnful to equrprnent the slgnal wlll be degraded rnterferenee m the l r e by ths eoherent rnterferenee Other voltage splkes lnclude llghmlng stnkes faulty equrprnent and so on These events oeeur very frequently see gure 3 28 Sam 3 m 2 2 Tm mlvuhage Suppress ID 2 3 Cuhermt lntafermee Induced ln Cuaxlal Cables by Magneue T r w Mm erreurt elernents sueh as vanstors or zener dlodes Currentrn the eenter eonduetor ean rnagneueally eouple to the shleld A varrstorrs avoltage dependent resrstor L Its purpose ls to eonduet agreatly lncreased eunent e away from other eornponents vvhen avoltag ls excesslvequot Use oftvvrsted pan eonduetors lnslde the shleld and slngle groundlng of A eornrnon devlce ls the rnetal oxlde vanstor MOV usedm surge max Sh e ds Ea m mm quot quot5 em proteetors Section 3 10 2 5 Use of a Longitudinal Choke or Neutralizing Transformer to Attenuate CommonMode Coherent Interference A toroidal ferrite magnetic core is used to reduce coherent interference caused by high frequency common mode interference pick up by wire pairs connecting devices Ferrite Toroidal Core The ferrite core effectively low pass filters any high frequency noise carried by the wires Section 3 10 2 7 Circuit Grounding Typical ground currents contain a large dc component on which rides a coherent ac component The common digital and analog circuit includes digital switching high frequency Good grounding practices include separate ground pathways for the digital and analog elements Sig l gr d5 lt Digital 1 Digital 2 Thus the high frequency noise from the quot digital circuits is more or less isolated from T 1 Hardware Egg smeld ground v M 15 Section 3 10 2 10 PhotoOptic Couplers A typical optocoupler consists of light to light signal coupling without any electrical conductivity between elements An input signal modulates the LED with shines on a photodiode The photodiode and amplifier circuit forms the output signal One use of optocouplers is isolated high speed serial digital data transmission Section 3 10 2 6 Experimental Verification of Cabling and Grounding Schemes to Achieve Minimum Noise Pickup Read this section on your own Section 3 10 2 8 Ferrite Beads and Feed through Capacitors An effective way of attenuating unwanted coherent interference on signal and power lines at frequencies above 1 MHz is with ferromagnetic beads strung on the wire carrying the offending interference These ferrite beads may be combined with a feed through capacitor and act as a secondorder low pass filter The name comes from the physical structure The exterior shield provides a dc pathway and the interior capacitor allows ac signals to pass through Section 3 10 3 Summary of Techniques for Coherent Noise Reduction The practice of the art and science of coherent noise reduction should begin with the elimination of local noise sources Offending noise sources can be isolated by grounded electrostatic shields Power leads must be lowpass ltered with the use ferrite beads and feed through capacitors Power leads should also be twisted in the supply wire to minimize magnetic coupling Ground loop issues can be addressed with proper separate ground pathways in the circuits elements Isolation transformers optocouplers mu metal and such are employed were useful 12 End of Chapter 3 Chapter 6 Survey of Sensor Input Mechanisms Section 6 3 Resistive Sensors Many sensors rely on a small change in resistance as the result of some external force Thus the resistance of the sensor varies around some baseline value R0 The most widely means of determining this small change in resistance is the Wheatstone bridge and other bridge variants It is important that the change in resistance AR be small Thus AR ltlt1 0 Section 6 3 l Resistive Temperature RTD Sensors Table 61 lists a few metals and their properties 39ITAEUE 1311 l39nil 39r39aia a rl quotgt 39 lLi ilI39iji39IF a UHFi3 lmJr RITE L HEUI a f qlr t39lai analmg Emmiswig igj l Ean ml39ifg Fli39 iaillinn i g iiiquot L39uii I if I w r m g I 393 I 1553 Hm all 2513 1 4 I Valium Hll l 39a ll lquot39lJl m a TE 2 5 ha in Lei H SEE Earall Sail 131 i All 5i I iiiJflui iarr El 39 If E 39JIE39JJ h 5 1 RTD resistance temperature detectors A voltage EMF is created when a metal is hot at one end and cool at the other end This property can be the intended effect thermocouple or a noise factor to be minimized RTDs Section 6 1 Introduction An input sensor or transducer is a device which permits the conversion of energy from one form into another form It is the first element in an instrumentation or measurement system For example it might convert a temperature into voltage The sensor s linearity range noise and dynamic response Determine the resolution sensitivity and bandwidth of the system Section 6 3 l Resistive Temperature Sensors In general the electrical resistivity of most metals increases with temperature The Bloch Griineisen formula not covered in this class shows that the thermal history the atomic structure and the specific atomic interactions of the metal all determine the resistivity versus temperature dependence The author gives this relation as a power series RT 25RaT 25 8T 252 Where 25 C is taken as a reference temperature for the material properties The temperature coefficient TEMPCO or is dRT dT 06 E E The author does not define the 2nd order coef cient B Section 6 3 l Resistive Temperature RTD Sensors Three methods to compensate lead resistance and temperature based EMF effects are A v RTD B RTD C RTD All three methods use a variable resistor that allows for the voltage at junction D to be compensated at some reference temperature Section 6 3 l Resistive Temperature Sensors Thermistor Thermistors are made from amorphous semiconductor material with some metal impurties One way to make this amorphous material is by sintering oxides sul des and silicates together Sintering is a process where powders of the various materials are heated but only below the material s melting points This heating causes the powder to adhere together The resistance of this semiconductor can be modeled by the Steinhart Hart equation 1 ABlnR C lnR 3 Tin Kelvin Where A B and C are coef cients usually given by the thermistor manufacturer in the data sheet Section 6 3 l Resistive Temperature Sensors Thermistor The thermistor can also be used to measure uid velocity If the power dissipation of the thermistor causes it to warm above the ambient temperature then a uid passing by the thermistor will cool it back down The power dissipation must be known and this approach allows for the change in resistance to be a measure of uid velocity Section 6 3 l Resistive Temperature Sensors Diodes I 2 1 ID revsat e Thus while the reverse saturation current will increase with temperature the forwardbiased current iD will decrease due to the exponential factor Solve for vD to discover that the diode voltage decreases about 2mV C for a number of common diodes n z 14 A Wheatstone bridge can be use to measure this change in voltage Section 6 3 l Resistive Temperature Sensors Thermistor Solve the last equation for the resistance R as a function of temperature to nd for a negative tempco NTC thermistor RT Roe 37 10 T is in Kelvin T0 is a reference temperature usually 298 K The coef cient B is a material property and must be given for the thermistor The tempco or for a NTC thermistor is dR 1 72 3 1 4000 For example let B 4000 a F 298 2 0045 Section 6 3 l Resistive Temperature Sensors Diodes The current in a forwardbiased diode is approximately VD W ID Irevsat 6 Where vD is the voltage across the diode and n is an ideality factor 1 S n S 2 The reverse saturation current also varies with temperature as T To l 0 The reference T0 is usually 300 K 2 IST ISat Section 6 3 2 Resistive Strain Gauges Resistive strain gauges fall into two broad categories bonded conducting lm cemented to the part under measurement unbonded a free to move system reacts to a force In the bonded strain gauges minute elongation or compression of the surface causes a small change in the resistance of the gauge The strain 8 resulting from the strain is given by AL in mm 8 m units 0f 0r cmd so on L m mm Let s the strain and Y the modulus of elasticity Then 8 1 LY Section 6 3 2 Resistive Strain Gauges The stress s is a measure of the work W applied to a speci c area A W s A In units of Pascals Pa in the SI system or psi in the English system The resistance of a given wire at a given temperature length cross sectional area and resistivity is RZLLQ A The author shows that the fractional change in R is given by AR AA Ap AL 2 R A 0 L Example Calculate the change in resistance AR for a strain 8 Steel wire diameter O 01 Young s modulus for steel Y 3 x 107 psi at 100 oF R 220 Q Gauge Factor GF 3 2 Load 1 lb The induced stress is s Load 2 11b 2 2psi l2732psi Area 7r 7r0 005 The strain caused by this stress is i 12732psi Y 3xlO7psi O 4244 x 10 3 424 4 microstrains Solve the gauge factor equation for the change in resistance m GF 2 3 AR 2 GFR8 3 22204244gtlt 10 6 2 O 2988 Q T Example Now plug in the values and nd the exact V0 AR 2 Q V0 V lt6Vgt 0 988 4R 2AR 4220 20 2988Q 6V3 393x104 2 0359x10 3V 2 0359 mV The approximate solution of V0 is AR 0 298 V0 z VS 6V 6V3 395x104 4R 4220o 2 0373x10 3V 2 0373 mV The difference between the two answers is O 07 The approximation appears to be quite good in this small AR case Section 6 3 2 Resistive Strain Gauges AR AA Ap AL R A 0 L The Gauge Factor is a data sheet value and is de ned by MApM AA Ap Ap AA M A L i 0 2 A L LA GF l lg geqn6ll L L L L L In units of Pascals Pa in the SI system or psi in the English system Poisson s ratio u is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force ML The Poisson39s ratio of a material in uences the speed of propagation and re ection of stress waves and thus a signi cant gure of merit Example Now nd the output voltage of a Wheatstone bridge under the previous conditions Resistors A B D 220 Q VS 6 Vdc The resistor X is the strain gauge X R AR R A E SRR S2R 2 VZV RAR V RAR 2V1R SRRAR S2RAR S2M Now nd the output voltage in terms of the input excitation VS M 21M M VOZV V IS 1R1VS R 21 2 2 22 22 Section 6 3 2 Resistive Strain Gauges The strain gauge is sensitive to temperature changes In our lab experiment the gauges are in an air conditioned room with very small changes in temperature and the effect will be ignored Read in the text about bonding on compensation arms to the bridge to reduce temperature based noise As was shown in the previous examples the bridge output voltage will be very small and require signi cant ampli cation The frequency response of bonded strain gauges depends on the mechanical system under measurement Typically dc to the audio range lt20kHz Unbonded strain gauges will not be covered Section 6 3 3 Photoconductors Photoconductors are materials whose resistance decrease upon illumination with light While vaguely similar to photodiodes or solar cells the semiconductor structure of a photoconductor only allows a change in the conductance upon illumination No carrier transport is directly enabled The active area which is illuminated is made very thin This enables good photon absorption in at least a speci c wavelength range Section 6 3 3 Photoconductors Photoconductor material and wavelength of their peak spectral I39GSpOIlSG TABLE 62 Properties of some photoconductors Wavelength of PC Bandgap Peak Response Rise time Material temp Energy eV AC um Fall time ZnS 360 0345 CdS 240 052 30 ms10 ms CdSe 180 069 15 ms15 ms CdTe 150 083 Si intrinsic 112 110 1 181 us Ge intrinsic 067 185 01 as 01 145 PbS 037 335 InAs 035 354 a 1 ns Te 033 775 PbTe 030 413 PbSe 027 458 2 us HngTe 77 K 50 5 us InSb 77K 018 690 GeCu 4 K 25 GeBe 3 K 55 Section 6 3 3 Photoconductor Example The dark conductance of a Si PC at room temperature with width 0 2 cm depth 0 001 cm length 0 02 cm Resistivity p 2300 Qcm n08 un 1350 cmZVs up 450 cmZVs IP 10394 sec The photoconductor is illuminated by photons with wavelength 7x 512 nm at a power of Pi 1 law The internal quantum efficiency 1 is the proportion of incident photons which are absorbed and contribute to the photocurrent The dark conductance is Area widthdepth 0 2em0 0010m D 3 43478x10 6o 0 Length 0 Length 2 3 x 10 Qem 0 02cm Section 6 3 3 Photoconductors Photoconductors can be made from a variety of semiconductor materials with light absorption targeted from the ultraviolet UV to the far infrared FIR wavelengths Cadmium Sulphide CdS Egap 242 eV visible range Germanium Ge Egap 067 eV near infrared range Indium Antimonide InASb Egap 018 eV far infrared range Mercury Cadmium Telluride HngTe far infrared range 2 22 um Liquid nitrogen cooled at 77 K www judsontechnologies commercadm htm Section 6 3 3 Photoconductors The total current is a photoconductor is PC VslGD GP VS the bias voltage across the photoconductor GD the conductance of the photoconductor under dark conditions GP the conductance due to light illumination The photoconductance GP is G Photocurrent ThOIeS p Rigel616ml U P V Length between electrodes2 he S The wavelength 7 in the above equation is valid up to cut off where the semiconductor bandgap no longer allows photon absorption for longer wavelengths Section 6 3 3 Photoconductor Example The photoconductance is GP 2 177205 ID5 L2 he Vs 1 602x10 19C0 810 4 sec1350 cmz 450 631 82 1x106W512gtlt109m 2x10 2cm2 6 626x10 34 Js3gtlt108 A L 57 672x10 182 576x1012 EZ z 6 VJ VJ V 1485gtlt104U I used the length of the photoconductor as 0 02 cm as given in the first part of the example and not the author s 0 01 cm in this part Section 6 3 3 Photoconductors Section 6 3 4 Conductive Relative Humidity Sensors This Opamp gives VO 0C Pi m gin Humidity sensors respond to the amount of water vapor in the air 391 VG One approach is the Dunmore sensor where two noble metals are p closely separated but not touching A thin hygroscopic coating of an ionizable salt covers the wires RL may be adjusted and null out the dark conductance Hygroscopy is the ability of a substance to attract water molecules from The total mean Square noise current input to Opamp is the surrounding env1ronment through either absorption or adsorption As the coating reversibly absorbs water vapor it will dissociate into rim 2 4kT GD GD GL GF B MS Amps ions which will conduct an electric current The process is kept reversible by using an low voltage low frequency ac excitation The MS noise increases with light power since GP increases with Pi Section 6 3 4 Conductive Relative Humidity Sensors Section 6 3 Another approach is through a variable capacitance mechanism Skip the following sections Water vapor is allowed to permeate the dielectric layer of a capacitor Section 635 6 36 and 637 element Typically the dielectric is a polymer material which may resemble a broom head to increase the absorbing area The capacitive value of the sensor can be measured by means of the various ac methods already examined Then this value is related to some known and calibrated value to determine the relative humidity Omega Engineering Model HX94C Temperature and RH sensor Section 6 4 Voltage Generating Sensors Section 6 4 l Thermocouples and Thermopiles Thermocouples are used to measure temperature A wide variety of physical systems generate an electromotive force EMF in response to some input Often thermocouples are designed to measure extreme temperatures such as in a combustion process l 100 to 1800 F When possible the sensor is designed so that the magnitude of the induced EMF is linearly proportional to the input quantity Type S top Very often the EMF is very small Signal conditioning including Type K bOttom amplification filtering and isolation is common A thermopile is a group of thermocouples in close proximity Section 6 4 l Thermocouples and Thermopiles A thermocouples consists of a junction between two dissimilar metal wires When such as system has a temperature difference between the two ends there will exist a small voltage Thus a basic thermocouple system must contain two couples one is at the desired measurement location the other junction is at a reference known temperature Section 6 4 l Thermocouples and Thermopiles The net EMF is a thermocouple system is NOT a linear function over the possible temperature range of the sensor The net EMF can be expressed as a power series 3AT2 CAT3 E0 AAT 3 The manufacturer might give you these constants A B and C More likely they will present a graph or tabular data which may be used to calculate the coefficients AT is the temperature difference above the reference junction temperature usually given as C The thermoelectric sensitivity of a thermocouple pair is de ned as dEO dT ST A BAT C AT2 usually in units of Table 6 4 Common Thermocouples Useful Temp Max Temp for Sensitivity in Materials Range C Short Periods uV C at C CopperConstantan 300 to 350 600 15 200 60 350 IronConstantan 200 to 800 1000 45 0O 57 750 PtPt90Rh10 0 to 1450 1700 0 138 5 0 12 1500 IronCopnic 200 to 860 1000 60 0 Chromel P Alumel 200 to 1200 350 40 55 between 250 and 10000C 39W95Re5W26Re7 0 to 2316 Section 6 4 l Thermocouples and Thermopiles Frequently a lead must be added to the thermocouple to bring the desired measurand to a more environmentally friendly location The addition of the lead usually copper to the system will have no effect on the thermocouple performance as long as the two junctions with the third metal are at the same reference temperature This property is called the Law of Intermediate Metals Table 6 3 Common Thermocouples Material ST vs Pt Material ST 223 Pt Bismuth 72 Silver 65 Constantan 35 Copper 65 Nickel 15 Gold 65 Potassium 9 Tungsten 75 Sodium 2 Cadmium 75 Platinum 0 Iron 185 Mercury 06 Nichrome 25 Carbon 3 Antimony 47 Aluminum 35 Germanium 300 Lead 4 Silicon 440 Tantalum 45 Tellurium 500 Rhodium 6 Selenium 900 ST vs Pt thermoelectric sensitivity versus Platinum baseline Section 6 4 2 Photovoltaic Cells Skip this section EEE l65 covers photovoltaics in very good detail See those lecture notes for any background information you desire Section 6 4 3 Piezoelectric Transducers Section 6 4 3 Piezoelectric Transducers Piezoelectric materials have the property that when strained a A piezoelectric transducer has mass mechanical stiffness and resulting open circuit EMF is created damping The word piezo is derived from the Greek piezez39n which means to Thus the transducer may have a mechanical resonance property This squeeze or press is exploited in sonar and other applications The corollary or converse piezoelectric effect is often also true If a voltage is applied across a piezoelectric crystal a strain is introduced However thls mlght be a tranSducer hmnatmn 1 3 5611501quot 31313110341011 Examples of the converse effect are ultrasonic cleaning devices piezo Piezoelectric material is also not a perfect dielectric Charge storage buzzers and audio transducers will leak at various rates and can affect a sensor s output Piezoelectric materials include Quartz Rochelle Salt sodium potassium tartrate tetrahydrate Tourmaline Topaz Cane sugar Section 6 4 3 Piezoelectric Transducers Section 6 4 3 Piezoelectric Transducers HF A piezoelectric transducer A uniformly applied force F will HF TOP area A compress the crystal by St it vo F orce i stress Area E T Thickness t Young39s modulus Y Stmm 5 The net charge displacement for the transducer is Let the crystal have a force exerted that compresses the thickness of the material Q 039 X F C0u10mbS Th 39 f h d 39 hl C 2 e capacnance 0 t 6 ans Ger 15 mug Y Where d 1s the charge sens1t1v1ty of the mater1al 1n Nm C 148180 F 442 X t The open circuit voltage across the transducer is The author calls the relative dielectric constant K rather than the more E0 2 Q 2 d X F common 8r C Agrg Section 6 4 3 Piezoelectric Transducers HF Section 6 4 3 Piezoelectric Transducers HF The author now de nes a voltage 1 Now clean up the units 5 i o o I T5t I sens1t1v1ty g as C F 601 Va F V0 am g E 14 d 2 g 2 N 442 442 442 442 Then FaradV F g m m2 md E0 ZgiorcetFarad Fm d g Aorcet VOHS rea m am rea 2 Farad 2 F This derivation works only if we assume all the numeric values for the Farad permittivity are lumped into voltage sensitivity g at the beginning 2 g Thus d m The pressure on the transducer active surface is FA Now we can rewr1te the open c1rcu1t voltage across the transducer as E Q dXF 8FaTWF0VC Forcet FaIad l j 0 2 E 1488 1488 g Area t t mereo ergo Section 6 4 3 Piezoelectric Transducers The crystal axes along which the piezoelectric material is cut determines the type of mechanical input to which it will have maximum sensitivity The transducer may be made to respond to compression shear forces twisting and bending V An electrical equivalent circuit is under the Cl assumption of frequencies well below the Fdi BL g 0 mechanical resonance 139 dQ ddFj Zak dl Olt l VI eq dz By Ohm s law in the s domain the open circuit voltage is ms 1 on dsF f0 2 Section 6 4 3 Piezoelectric Transducers V Olt The author now determines the transfer Fd BL 0T7 function for the output voltage versus the applied force Voltsgt 85 H S1C The transfer function is in the form of high pass lter Sk sac H high pass Once again I come up with a different result than the author for eqn 642 It appears to be typo errors Section 6 4 3 Piezoelectric Transducers 3pm Use this circuit to amplify the CL fWV V transducer signal 4 L A em 4 l Assume the input impedance into the opamp is very high Also assume the opamp is ideal With the virtual short at the opamp input the current from the transducer must ow in the feedback elements By node analysis 0 V V 1q 00 gt 0 qu gtV0 1zF z 2 q RF V0 4 R 1 4 S 1 dsF 1 RFCF S RFCF Section 6 4 3 Piezoelectric Transducers If we assume the initial force on the piezoelectric material is zero then a 31 0 i Olt V0 S 61 SF Zeq The parallel impedance Z 01 is L L 1 Z L 1 RL ScT SRLCT l S RLCT With this result the output voltage V0 is 1 V0sdsF C71 SI RLCT Section 6 4 3 Piezoelectric Transducers gt DC response for a piezoelectric transducer is Fd BL 0T7 not possible due to charge leakage l The leakage conductance 1s GL 2 R F s 1 L CT is the capacitance of the transducer plus all the leads and device connected to it Typical CT is on the order of lnF Section 6 4 3 Piezoelectric Transducers Again de ning a transfer function of the output voltage of the opamp versus PM BL the input force we have CLL 0 T EOA VOltSgt Se F s H RF CF This section seems hopelessly lost with typos The author s discussion follows my equations rather than his own Example Example Use the following numeric values T Ci TNT The transfer function is given by T Ci RF 1010 Q Ed 51 EOA J Si 30 b 51 EOA CF5xlO399F5nF T HSV0S CF1 T d223X1012CN 2 F Sm 1 The time constant T is The phasor form is L MC 239 RFCF 1 olooxs x10 9F 50sec 4 RF CF V F ja The 0011161quot frequency 0c 13 For high frequency yet still below mechanical resonance frequencies 1 1 a 002racy d 23x10 12i c 50 360 H 39a z N 46X104L sec J CF 2x10 9F N The comer frequency fc in Hertz is d ca 002 3 I 3 1 8 X HZ 27 27 rad Section 6 4 4 Pyroelectric Sensors Section 6 4 Sensors Whose Voltage Output is Proportional to ddt A voltage will be induced in a coil of N turns when it is surrounded Skip this section by a time varying magnetic ux I d di E0 N L dt dt Many input transducers use this principle for example dynamic microphones accelerometers tachometers and more Skip 6451 Section 6 4 5 2 Electrodynamic Accelerometer u u Section 6 4 5 2 Electrodynamic Accelerometer u u Case gt Case The gure shows a linear accelerometer 3quotquot g 5 g if The motion of the mass is given by 3pm J Coil quot Coil The permanent magnet is movable ll M H B o o k 0 ll 1ns1de a c011 xed to the accelerometer s g xm xm x6 30639 x6 g base 1 F l 1 F l o o o o o o o The input is the acceleration of the case The magnetlc mass 1s constrained by a linear spr1ng w1th st1ffness ks The author Laplace transforms the above equation and after some The mass is also surrounded by a viscous oil which damps the motion algebra nds of the mass M Sk VoS 3505 2 The output voltage V0 1s proportlonal to the relat1ve veloc1ty between S the case and the moving mass and is given by M B ESE1 Under a constant acceleration the mass will experience no force V0 KvOCC km separate from the case Thus there is an output voltage only under changing acceleration Section 6 4 5 3 linear VelocitySensors Section 6 4 a 2 Hall Effect Sensors charges moving in the presence of a magnetic iieldwill experience a force Skip this section Fqvx8 qisthe char e v is the velocity at which the charges are moving Bis the magnetic iieldvector The magnetic forces will cause a charge density to build up until an Lorentz magnetic force 5 vwsy Section 6 4 a 2 Hall Effect Sensors Semun 4 6 Z Hall Effect Sensurs The average dn39 velocity of electrons is The author de nes the Hall coef cient as 71 7 for electrons qn quot c r r Where 1X is the cunent density in the semiconductor bar 7 n is the electron doping density With this coef cient the Hall Voltage is RHIB The resulting electrostatic eld E1 is V h Volts 71x5 Val E T Hall effect sensors can measure an ac cunent and infer a power ow ifthe ac voltage is known The Hall voltage see the geometry above is Hall effect sensors can also measure magnetic fields and thus be used 15 Volts h is the height ofthe semiconductor 39 W slab as gate counter where a magnet passes a pickup point Section 5 s Sensors Based on Vanahle Magnetic Coupling Section 6 51 The LVDT me rinea 39 39 39 14 u r 39 L r r u widely used mechanical input sensors and output windings witrs are used to measure linear mechanical displacement or positi on Excitation must be with an ac signal LVDTs can indirectly measure force pressure acceleration or any quantity that causes a linear displacement This process can both measure or generate linear or rotational displacements depending on the physical structure ofthe device www sensorland com A very good white paper on LVDT with signal conditioning can be found at htt focusti com cncnlltans ra946s ra94o df seenonosl Thewm type of eleetneal transformer whlch has three e LV solenoldal cells plaeed endrtorend around a tub The eenter eorl ls the pnmary anolthe two outer eorls are the seeonolanes m an arrangement ealleol aMaxweH Pm www nrgerrnslmnsnts eonmonuth cyllndncal ferromagneue eore attaeheolto the object whose posruon ls to be measured sholes along the axls ofthe tube seenonosl Thewm When the eore ls m rts eentral posrtron equrdlstant between the two seeonolary eorls equal but opposrte voltages are mducedm these two eorls the ou voltage ls zero When the eore ls dlsplacedm one dlrecnon the voltage m one col rnereases as the other oleereases eausrng the outputvoltage to mcrease from zero to axlmum Take thrs dlrecnon as the ease where the voltage ls m phase wrth the pnmary voltage e eore moves m the other ohreeuon the output voltage also rnereases from zero to amammum but rts phase ls opposrte to that of the pnmary Seman SZ Synchmsandstalvexs Sklp ths seeuon Seenonosl Th2 var mt An ae eurrentrs olnven through the I p mary causmg avoltage to be 39 mducedm eaeh seeonolary Hg m proporu onal to rts mutual quot r rnoluetanee wth the pnmary 4 l we The frequency of the ae exertatron ls usually m the range 1 to 10 kHz 0 to 20 kHz also useol on oeeasron As the eore moves the mutual rnoluetanees ofthese three eorls ehanges eausrng the voltages mducedm the seeonolary eorls to ehange The eorls are eonneeteolrn reverse senes so that the output voltage ls the dlfference henee dlfferentlal between the two seeonolary voltages Seenonosl Th2 var The magnrtuole ofthe outputvoltage ls proporuonal to the dlstance rn veol by J M the eore up to rts hmrt oftravel whlch quotlquot D ls why the demos ls olesenbeol as mi lmear NW The phase ofthe voltage rndleates the duectlon of the ohsplaeement Note thatrn the gure above the author has rneluoleol the dot Thus the dlrecnon ofeurrent anol polanty ean also be oletermrneol by the clot eonventron en e eore moves m the other dlrectlon the outputvoltage also rnereases from zero to amammum but rts phase ls opposrte to that of the pnmary Seconn Vamblz Capcxlnnn Sensuxs Sklp thrs seetron Seeuon 6 7 Fiber Opue Sensors Cover as time permits Section 6 7 1 MagnetorOpu Current Sensors Seem s 7 2 Measurement of Optical Rotation Section 6 7 3 Fiber Optic FO Mechanosensors Skip this section aaaa Seeuon 6 8 Photomultlpller Tubes Skip this section Section 6 10 Electrorchemlcal Sensors Electrochemical sensors operate in a Wet environment in Which the substance to be measured is dissolved in Water or another solvent The author brie y covers sensors Which rely on a system of halfcell potentials such as a pH measuring device in section 6101 The chemistry is interesting but notice that it is the ampli cation of veryhigh r4 410W r L In section 6102 polarographic methods are discussed Which use a signal to drive an electrochemical reaction that can be measured usually as a response time to a steady state Section 6 11 Electronic Noses Skip this section Section 6 9 Ionizing Radlahon Sensors Skip this section Section 610 3 Fuel Cell Eleetxodes Then the author brie y looks at a sensor that measures the rate at Which some reactant is depleted This is the lel cell electrode method use ll in glucose studies Read section 610 as background material This Wet chemistry topic currently receives a strong effort by UC Davis for example As the various biomedical initiatives in the region take root this topic may receive even more attention Section 612 MechanorOptzc Sensors MechanoOptical sensors are transducers Where a mechanical parameter such as di ement or velocity causes changes in the intensity of transmitted or re ected light The intensity changes may be due to direct in uences such as an opening changing size and admitting more or less light pass through Think of venetian blinds The intensity changes may also be due to indirect in uences such as polarization states or optical interference The simplest sensor is Where a sha angle is sensed With an optical encoder disk Velocity is the derivative of the position signal Seettan 5 l2 l Opncal Codtng Dsks Optleal codmg dsks are usedto eonvert analog shaftrotanon angle to a dgttal word proporuonal to the angle Two broad types Incremental Pdsmtm En nseesu a ehange m po ED slu on but dsts e the sensor must be homed to provlde a frame ofreferenee Absdzum Pdsmdn Encoders e as the gure to the nght shows resolves t o fthed the eombrnauon ofnngs o a nlte posmon forthe entrre evlee Seetmno l2 3 LnserDopplerVeloermetry computedhue lnerementunptteu encade Absolute opted encade Laser Doppler veloelmetry LDV prondes a nonrcontact method of e es measunng the llnear veloelty of ulds and gas p Laser hght shown D m ovlng Let a slnusoldal wave 1 dlstanee dto heheoptertarget onto amonng object wlll expenenc oppler shllt m Veloclty dependmg on whether away or towards the observer arul e the famlllar ls the objectls eave the transmltter CTRX at veloelty e over a T The helreopter ls movmg towards the recelver 1200 at aVeloclty V as shown above Seetmno l2 3 LnserDopplerVeloermetry The speed ofllght wavelength and frequency are related A by 11 Thus the phase ean be wntten as In the target was not movmg we wouldbe done Buttargetrsmovmg a e towards the reeelver forthe frequency at the recelver we h v 7 d mat y dt H2 man I 1 gnaw 2 dt 27 Seetmno l2 2 Sagmc Effecl Sensrng afAnguhrVelncny Sagnae effect gyroseopes are the state ofthe artposltlomng system 5 slgnal ls eonstantly requlred andthe sensor wlll workln the presenee oflarge amounts of nolse m all envrronments sueh as outer spaee underwater and underground The deVlce must start from aknown reference pomt andlnfrequently be resetto overeome system errors However the eosts of these mostly handrmade deVlces ls very hlgh andlarge seale rmplementauon has not oeeurred Read ths seeuon and seetron 7 2 2 2 as mterestmg background lnformatlon Seetmnolzz useanpperVelaemetry h Veloclty V ean be resolved rnto parallel and perpendlcular eomponents respeetlve the TRX e T lme The veloerty parallel to the hne ls V eos e The veloelty perpendlcular to the lme ls V sm a war ch senses the V eos e slgna1the other eomponents re eet elsewhere Thus the MK output ean be wntten as t e Eanmntu The phase angle luls the phase lag between the transmltted slgnal and the reeelved slgnal 2 z Seetmnolzz useanpperVelaemetry ennhaiyeosa The Doppler frequency ls the amount of frequency shllt zmnchas A typlcal LDV geometry ls ths gure In a slmllar denvauon and atnonrrelatlvlstl Velocmes V ltlt e Z V eosaeeosn Hz 1 flaw The authorbne y dlscusses how exqursltely sensluve thrs deVlce ean be Section 6 12 3 LaserDopplerVelocimetry Uses of Laser Doppler Velocimetry include studies of Wind shear clear air turbulence mmadoes End of Chapter 6 automotive streamlining respiratory physiology blood ow aircra design combustion dynamics


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