Essentials Applied Elec Engr
Essentials Applied Elec Engr TMAE 5132G
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TMAE 5132C Essentials of Applied Electrical Engineering Georgia Southern University Mechanical and Electrical Engineering Technology Spring 2009 Measurement Systems Prof AlbaFlores METEET Dept Measurement Systems Generalized Measurement System 3 basic stages TransducersSensors Overview Hysteresis in sensors Generalized Measurement System Three stages of a Measurement System 1 SensorTransducer 2 Signal Conditioner 3 ReadoutRecording Generalized Measuremen System 3 stages of a Measurement System Physical Parameter Sensor Signal Readout Transducer Conditioning Recording Three stages of a Measurement System 1 Sensors I ansducers Physical Parameter Sensor Signal Readout Transducer Conditioning Recording gt Transducers Transducer are devices that change one form of energy into another form usually to electrical One Classification of Transducers 1 Modifiers 2 SelfGeneratinq Self excitinq 3 Modulator Classification of Transducers 1 Modifiers energy is modified not converted The form of energy will be the same at the input and output eg spindle displacement to displacement 2 SelfGenerating Self excitinq Energy input is converted to new form at the output usually a very low electrical quantity No external power is required eg thermocouple temperature to electric potential 3 Modulator External power is required Energy is converted to electrical energy at the output eg Linear Velocity Transducer displacement to voltage Examples of Transducers Pendulum gravitational acceleration to frequency orpenod Liquid column pressure to displacement Thermocouple temperature to electric potential Pressure Thermometer temperature to pressure Photovoltaic light to voltage Piezoelectric displacement to voltage or voltage to displacement Transducers cont Transducers can also be classified as Input Transducers Sensors Convert I hsical I arameters such as tem erature pressure force magnetic field strength etc into electrical energy Output Transducer Actuators Convert electrical pneumatic hydraulic or other forms of energy to mechanical force or CllSplaCemenL eg speaker Examples of Transducers Temperature 0 Thermocouple AT gtAV Temperature to electric potential 0 ThermistorRTD Resistance Temperature Detectors AT gt A9 Tem erature to resistance change Photoelectric o Photodiode ALight gt A1 Light intensity to current Examples of Transducers ForcePressure 0 Strain Gage AF gt A9 Stress to changes in resistance Piezoelectric ADispl gt AV Displacement to voltage andor voltage to displacement Acceleration 0 Accelerometer Aaccel gt AC or AV Types of Transducers cont PosMon o LVDTAdisplacement gt DC orAC AV Linear Variable Differential Transformer By moving a magnet through a coil the transducer will output a voltage proportional to the instantaneous velocity 0 Linear Potentiometer Adisplacement gt A9 Velocity o LVT Avelocity gt DC or AC AV linear velocity transducer converts linear mechanical motion to electrical signal Transducers summarn Physical Paramete39 Electrical Quantity Temperature AV Voltage Displacement AQ Resistance Velocity AI Current Acceleration AQ Charge Force Hysteresis in Sensors Hysteresis in sensors Hysteresis is a measurement of an electrical component39s change in performance as a result of continued operation When a component is operated for the first time for example a temperature sensor reading temperature from cool to warm it will follow a particular curve When you then move the s stem in the o osite 7 direction say change the 339 temperature back to cool the component will follow a similar curve back to its original state but slightly displaced from the l original curve voltage temperature Example of Hysteresis Output Value Hysteresis 2 4 Input Value 6 10 Three stages of a Measurement System 2 Signal Conditioner Physical Parameter Sensor Signa Readout Transducer Recording Conditioning Signal Conditioning Definition Signal conditioning is the process of interfacing to a sensor amplifying and filtering to make the signal ready for display or analysis 0 There are many possible functions of the signal conditioning stage 0 We can have multiple signalconditioning sections in a specific measurement system Most Common Signal Conditioners Amplification Attenuation Filtering Differentiation Integration Linearization Combining a measured signal with a reference signal Converting a resistance to a voltage signal Converting a current signal to a voltage signal Converting a voltage signal to a current signal Converting a frequency Signal ID a VUIIage Signal Three stages of a Measurement System 3 Readout Recording Physical Parameter Sensor Signal Readout Transducer Conditioning Recording TIVIAE 5132G Essentials of Applied Electrical Engineering Georgia Southern University Mechanical and Electrical Engineering Technology Spring 2009 Lecture 1 Electrical Energy Sources Prof AlbarFlores METEET Dept 1132009 The Nature of Matter Matter is anything that occupies space or has mass Smallest parts I Element purest form I Compound two or more elements are mixed together The Nature of Matter quotmm mm Example of Elements Example of compounds 1132009 Examples of conductors semiconductors and insulators Semiconductors Copper SlilCOn Bakelite lron oermanrum Glass teel Gallium Arsenide Mica Aluminu Porcelain Silver and Tin Paper Rubber Air Static Electri ity Applications Assignment due Wednesday Jan 14 2003 Part 1 tookror applications or static electricity and write a Hi this Pr 39 couple or paragraphs abo rd the reference S and include a briefintmduction to the concept of Static electricity eg Electrostatic precipitator it is used in industry to reduce air pollution Basic Electrical Circuits Resistance voltage current Ohms Law and Power Se r39les Clrcults Parallel Clrcults Electric resistance a a material39s opposition totherlow or electric current measured in ohms Electrical tension or voltage is the dirrerence orelectrical potential between two points oran electrical or electronic circuit expressed in volts Electrical Current is a ow or electrical charge carriers usually electrons or holes electmnrde cient atoms the standard unit is the ampere R 1K Ohm39s Law Ohm39s taw defines the relationships between V voltage 1 current and R resistance On mistheresistanceval r eam e oh ue through which one volt will maintain a current o on pere VlxR Power in Electrical Circuits Electrical Power P in a circuit is the amount ore is absorbed or produced within the circuit The un measurement is thewatt w nergythat it or A source orenergy such as a voltage will produce or deliver power whilethe connected load absorbs it P is the product orvoltage multiplied bythe current P V x I ing Ohm39s law and substitutingror V I and Rthe d By us formula for electrical power can be roun as P V2 R P F x R TEMAE 5132 Electric Devices and Measurements Georgia Southern University Mechanical and Electrical Engineering Technology Spring Semester 2009 Passive Filters and RLC Circuits Prof AlbaFlores METEET Dept General Aspects of Signal Filtering Filters are used to remove unwanted inputs AKA Electrical Noise Filtering is the process of attenuating reducing unwanted frequency components of a measurand while permitting the desired fre uenc com onents to I ass Filters Ti rmi Corner Fre uenc fC Passband Stopband Lowpass Filter Highpass Filter Bandpass Filter Bandstop Filter Notch Filter Filters Types Butterworth Filter Chebyshev Filter Bessel Filter We will cover two types of filters 1 Lowpass Butterworth passiveRC 2 Highpass Butterworth passiveRC Passive lters Filters are circuits that are designed to allow signals of certain frequencies to pass to a load and to reject all other frequencies Filters that allow frequencies below a certain cutoff frequency fc to pass and reject all other are know as lowpass lters Filters that allow frequencies above a certain cutoff frequency fc to pass and reject all other are known as highpass lters Vin Filter Vom Figure 1 Block Diagram ofa Filter Vout Vin Vout Vin 1 0 0707 Frequency passband stopband 1 11 m c m radsee fc E Stopband Pass band cutoff frequency frequency Frequency response of a rst order luwpass lter Frequency response of a firstrorder highpass lter Figure 2 Typical Frequency Response of Filter Lowpass Filters Because capacitors and inductors respond to frequency they are used to design and construct lters Avery simple lowpass lter may be constructed f V a ss no or RL circuit as shown in gure 3 21 RC low pass filter 393 RL low pass filter Figure 3 Passive First order lowpass filters Note that in a RC lowpass lter because XC 1275fC then as frequency decrease XC increases This means that the capacitive reactance approaches to an open circuit for low frequencies and Vout Vin R Vin JWT Vout 1 0 E I C 1 r c ZnRC Vout Vin 1n vv EW Frequency RC low pass lter p quot f 5 p Hz C cutoff frequency Frequency response Ufa rstorder lowpass lter Figure 4 RC Iowpass filter Note that as frequency increases XC decreases and for high frequencies the capacitive reactance approaches a short circuit and Vout 0 The frequency response of a low pass filter can be represented as Vout Vin 1n uv H E Frequency passband stopband fc cutoff frequency Frequency response of a rstorder lowpass lter Figure 5 Frequency response ofa Iowpass filters The cutoff frequency The cutoff frequency is the frequency at which the output voltage drops in magnitude to 7071 of its maximum value It is also regarded as the frequency at which the power dissipated in a circuit is half of its maXimum value vout vin 07071 For the RL lowpass filter XL 03L therefore as the frequency decreases XL decreases then the inductive reactance approaches to a short circuit and Vout z Vin Note that as the frequency increases XL increases and the inductive reactance approaches to an open circuit and Vout z 0 The cutoff frequency is at 03C R L At this point the output voltage vout 0707 vin Vin 10 L V v v 1 Vout m R c L R ft R En Frequency 275 L passband stopband fc cutoff RL lowpass lter frequency Frequency response nfa rstorder lowpass lter Figure 6 RL lowpass lter Hivh ass Filters A very simple highpass filters may be constructed from a series RC or RL circuit as shown in figure C Vin quot i Vout Vin R V w L out C RC R w 3 L c L rc 1 77ERC c 7 R 27EL RC highpass lter RL highpass lter Figure 7 Passive First order highpass filters Note that in a RC highpass filter because XC 127ch then as frequency decrease XC increases This means that the capacitive reactance approaches to an open circuit for low frequencies and Vout z 0 Vout Vin 10 C KIli quotVout u7u7 RC R RC Stopballd c Passbaud quotI Q fr eeeee cy Figure 8 RC highpass filter Note that as frequency increases XC decreases and for high frequencies the capacitive reactance approaches a short circuit and Vout 2 Vin For the RL highpass filter XL coL therefore as the frequency decreases XL decreases then the inductive reactance approaches to a short circuit and Vout z 0 Note that as the frequency increases XL increases and the inductive reactance approaches to an open circuit and Vout z Vin The cutoff frequency is at too R L At this point the output voltage Vout 0707 Vin x I 30 wa 9 m m 7 L ciltoff gt RL highpass lter eq39 Frequency response of a i39strorder highpass ner Figure 9 RL highpass lter Resonant or Tuned Circuits Bandpass and Bandstop Filters Series Resonant Circuit RLC circuit When a RLC circuit is energized at a frequency at which the inductive reactance is equal to the capacitive reactance XL XC it is said that the circuit is at resonance This particular frequency i alled W resonant frequency f0 of the RLC circuit IS gt R XL21rfL f 1 0159 L 1 21 ii C i c quot5 Y5 ijc 21rfC a Figure 10 RLC resonant circuit The impedance of an RLC series circuit is given by z R2XL XC2 Note that at resonance we have The frequency response of a series tuned circuit appears as a bellshaped curve Z ohms frequency frequency HZ Figure 11 Frequency response of a RLC resonant circuit I Note that the impedance curve shows a minimum value at the resonant frequency I Note that the current curve shows a maximum value at the resonant frequency Quality factor of Tuned Circuits The Q factor indicates the sharpness of the bellshaped frequency response of a RLC tuned circuit Amplitude Ql least selectivity Q3 medium selectivity Q3 greatest selectivity frequency HZ Summary of of resonant RLC circuits Parallel circuit Characteristic Series circuit 1 l Resonant e uenc wt 7 q y xLC JLC no L 1 R C Qnahty factor Q R or woRC wOL or we E Bandwndth BW Q Q Lower and upper cutotf 1 2 m 1 i 2 E frequencies 11402 0 1 i a quot 0 2 2 BW ForQ210wwz wot out Passive RC filters Practical Design Equations 1 Lowpass Butterworth passiveRC 2 Highpass Butterworth passiveRC Filter Desi n E uations Lowpass Filter Equations f I rimar fre uenc fC 2139rRC 1 fC cutoff frequency 3 1 1 ffc2n12 n filter order D tan1ffc Phase Lag R O vwx O Vin lt C Vo Filter Desi n E uations Highpass Filter Equations f I rimar fre uenc Fc ZWRC 391 fC cutoff frequency G ffCI 1 ffc2n12 n filter order D 90 tan391ffc Phase Lead Filter Example 1 Lowpass Filter Example Whv15rrRLwPiritwith R1 K0 and C1pF The input signal to the amplifier has a primary frequency of 10 Hz and amplitude of 5V The noise component has frequency of 200 Hz and an amplitude of 1V Determine the equation of the output signal R OHM 0 Vin l Filter Example 1 Amplitude W I A vow ham I I 00 CD Unfiltered Filtered Low Pass Filter Example 01 02 03 04 05 Time sec Filter Example 2 Highpass Filter Example We have a 1St Order RC HighPass circuit with R39KQ and C001pF The input signal to the amplifier has a primary frequency of 1000 z and am V e noise component has frequency of 60 Hz and an amplitude of 2V Determine the equation of the output signal 9 O R 0 Vin R VO TMAE 5132C Essentials oprplied Electrical Engineering Georgia Southern Universit Mechanical and Electrical Engineering Technology Spring 2009 Force Measuring System Strain Gages Review Prof AlbaFlores METEET Dept I E modulus of elasticity Young39s modulus v Poisson39s ratio and G modulus of rig39dity Shea modulus 39 u uucu uuu i a 39 39 g 39 39 A uniaxial sizess slate is used to de ne the constants E and V While pure shear is used to de ne G H P M Figural Fmssun s Rauu the geuemlized Hooke s law Speumen W r u uu u u uau L material L t 39 M an E2 1 v G determine Poisson39s ratio ow the tensile test 7 V V r r v m 1 5 it A 4 my u u uniaxial stress state 4222009 Strain Gages I Measuring Strain When a force is applied to a structure the components of the structure change slightly in their dimensions and are said to be strained Devices to measure these small changes in dimension are called strain gages StressStrain Relationship For elastic materials stresses in a single direction uniaxial stress the strain is related to the stress by Hooke s Law E IaEa o Stress psi E Modulus of Elasticity or Young s Modulus psi 5 Strain inin or ustrain Strain Gages 0 Electrical Resistance Strain Gage The electrical resistance strain gage is a common device used to measure strain in structures and also as a sensing element in a wide variety of transducers including those used to measure force acceleration and pressure R pLA p Resistivity 0 cm L Length of the conductor cm A Cross sectional area of the conductor cmz It can be shown that Ea 1GrlURRl Gf Gage Factor typically around 21 for wiretype gages R Unstrained gage resistance 0 AR Change in resistance due to strained conditions 0 4222009 Wheatstone Bridge Circuit TworElement Varylng Bridge SinglerEiemem Varying Bridge oppOSi e reg Wheatstone Bridge Circuit TworElementV rying Bridge same ieg FourrEIement Varylng Bridge 4222009 4222009 Strain Gages Bridge Output Voltage In terms of bridge resistors Va VER2R1 R2quot R4R3 R4ll In terms of change in resistance Vu VE4 ARR AR2 single element Vu VEZ ARR two element same leg Vu VEZ ARR AR2 two element opposite leg Vu VE ARR four element In terms of strain from mechanical setup Vu VEGf 51 52 53 54 4 Force Measuring System Application 1 One strain gage is mounted on a piece ofaluminum as shown below with a nominal resistance of350 Q and a gage factor of 21 The member is subjected to a 5000 lb axial force The piece of aluminum is 12quot x12 and the modulus of elasticity is 10x106 psi for aluminum Calculate the output voltage if the gage is connected into a single element Wheatstone bridge at 5V 0 FA 5000 b12 x 2 20000 psi Axial Force a oE 20000 psi10x106psi 0002 inin or 2000 ustrain v0 VEGf214 SV21 0002 4 525 mV Force Measuring System Application 2 T below with a nominal resistance of 350 Q and a gage factor of21 The cantilever beam is 6quot long by 1quot wide and 316quot thick and the modulus ofeiasticity is 10x106 psi for aluminum Calculate the output voltage fora 10 lb input force ifthe gage is connected into a singleelement Wheatstohe bridge at 5V 0 Myl Fiexure Equation F Drille Strain Gages FL v h2 i 1112hhi o 6FLbh2l So 6 6FLEbh2l s 6l10ibl 6quotl10x105l1quot 018752 0001024ihih or 1024 Stain VD VEGEjl4 5Vl21l40001024 0001024 538 mV Strain Gages 0 Strain Gage Orientation 0 Axial Strain elongation 0 Lateral Strain proportional to Poisson39s ratio v ELEa EL Lateral or transverse strain Ea Axial strain Axial Gage Lateral Gage Orientation Orientation Axial Force 4222009 Strain Gages I Axial amp Lateral Gages Lateral Gage R VAR Rl Axial Gage R AR N V0 VglleRl Rzll R4R3 R4 R1 RVAR R2 RAR R3 R4 R R2 R4 v0 VgllRARRVAR RAR RZR TwurElement Varying Brldge a leg m v0 VE4AR1vR Axial amp Lateral Gage Example 1 Two strain gages one measuring axial strain and the other measuring lateral strain are configured in a half bridge circuit The strain gages are mounted to a V2quot x V2quot square piece of metal with a modulus of elasticity of 12 x 106 psi and a poisson39s ratio of 029 The strain gages used are 120 Q with a gage factor of 21 and the applied voltage to the bridge is 5V Determine the tension force in bs that the bar would have if 4 mV were measured on the bridge output TworElememVarylng Brldge same lea Axial Force v0 VEll AR1vR or AR AVORVE1V AR 445mV12005V129 03349 0 ea 1GFARR 1210334901200 1329 ustrain o FA and E 05 so 5 oE FEA F EEA 1329 ustrain12 x 105psi A x 1Aquot F 3987 lbs 4222009 TMAE 5132C Essentials of Applied Electrical Engineering Georgia Southern University Mechanical and Electrical Engineering Technology Spring 2009 Complex numbers RLC circuits with AC signals Prof AlbaFlores METEET Dept Complex Numbers Review Review Section 93 of Alexander The Complex Xy Plane Rectangular form Any complex number 2 can be uniquely represented as a point in the xy plane where the xcoordinate is Re 2 and the ycoordinate is Im 2 111m ginaly mu m Real zxjy Polar form We can also write z in polar form as zrefercosejrsin 9 Where r and 9 are real and equal to the length and angle of the vector Inmginary cosEl l T L11PII r v I I 1 f smiEl AIM x F P II 1 ff f as Kiri 6quot The complex conjugate of z r 619 The magnitude is z sqrtz 2 r Note that 1re 191r eije Raciangular Farm Z X y Polar form 3 r EKp emial Farm 2 r g Multiplying Complex Numbers Multiplication is distributive ajb cjdac bdjadbc In polar form we multiply the r 39s and add the 9 395 Example 1 if 21 r1 6191 and 22 r2 6192 Then 3991 92 Dividing Complex Numbers Example 2 if 21 r1 6191 and 22 r2 6192 Then 21 z2 r1r2ef91 92 Euler39s Theorem 19 0059 sin 9 ej27r1 61002 1 ef 2j cosGefe e 13992 sineefe eJG2J RLC Circuits with AC Signals Review Sections 94 95 96 97 Inductors and Capacitors are electrical devices that can store and release energy Capacitors store energy as electrical charges electrons lack of electrons storage voltage Inductors coils storage energy as magnetic fields storage current capacitanceof a parallel plate capacitor 24 C a E A is the surface area of each plate d is the distance between the plates E is the permittivity of the dielectric material between the plates Dielecuic with W l l p 1 Metal plates each with area current voltage rella ti uln hip uf the capacit r I f I C d vuvllli zage curren t reliati n Hf the capacitur t t V I39d nr V ll39fdt 1313 El Ell391 A capacitor is an open circuit to dc The voltage an the capacitor must be cantiinuaus The linkage an a capacitor cannot charge ahrup ly The capacitar resists an abrupt change in the voltage across it IN DUCT 0 RS An inductor is a passive element designed to store energy in its magnetic field Any conductor of electric current has inductive properties and be regard as an inductor But in order to enhance the inductive effect a practical Inductor is usually formed into a cylindrical coil with many turn of conducting wire Length Crosssectional area A Core maten39al Number of turns Ni Inductance is the property whereby an inductor exhibits opposition to the charge of current owing through it measured in Henry H N is the number of turns I is the len A is the crossasectional area M is the permeability of the core The care may be made of iron steel plastic or air Length UHJ Crosssectional area A Core maten39al Number of turns Ni Thevuttage across the tnductnr E v L Lit L t5 the inductance of the inductor the unit of tnductance is the Hean H The currentveilitage relationship DCI t I 139 I MIMI er f J1 tdtft An inductor acts like a short circuit to dc An important property of the inductor is its opposition to the change in current owing through it 1ihe current through an inductor cannot change instantaneously an inductor opposes an abrupt change in the current through it Component Behaviors Resistor 1mm luumiu phn of mum and Ullrlg r39 m a misrm Current and voltage are in phase Inductor Time c lunmiu plulj If c39111391 t 11r and Md 2111 fur an iurlut39mt39 HI Current lags the voltage by 90 Capacitor rum luumm pm quot1 Huu39ur and mle rm 1 Wm Current leads the voltage by 90 7 LI In Tm dmmxiu rhr huhan ur imImmm rmismrm mu rnpm39imm im mmiwly V 11 7 071 1 0R Thu impedance Z and admittance Y of a cumpoumr arr du nml hy Ohm s Law 1 7 l 139 mul Z H139m39i1n39twall l plt M39uIminuxI th mmt Mu x39al lummr u Lcn39ur39ml 39 and Z un x39umplm alum Z11EJA39 l m vqniwh ull 7 14313 7 NZ inlpeda mce Z admittance lquot 111 13quot 11quot139HL 39139 and Z Hl39fquot liilllll3931 M39HIHI H JI39 quotIt llli39i39irilt lllLT fa I ESiEt l lEC 3quot reactanm If canductance H sua39mptance Impedance VM VM Z 7 1Mamp 1M L 939 219 2a Rw jXw Ru resistive component XW ReaCtance Component Series Equivalent Impedance Parallel Equivalent Impedance Element Impedance R Z R L z ij sz90 1 1 C Z L 90 JmC 03C 03C ZRjX rectangular form Z Z 192 polarform Example 3 Determine the equivalent Impedance of the TOIIOWIng RLC series circuit if i f 100 Hz and ii f 400 Hz Consider the voltage source vt 25 cos 0 t 45 b Calculate the current for both frequencies i0 Assume R R 25 Q L 20 mH 1 0 L b DU uF T C TMAE 5132G Essentials of Applied Electrical Engineering Georgia Southern Universit Mechanical and Electrical Engineering Technolog Spring 2009 Linearity Errors and InputOutput Loading Errors in Measuring Systems Prof AlbaFlores METEET Dept Lecture Content Decibel definition Input amp Output Loading Errors Linearity Errors An Example Force Measuring System Application 0 Signal Conditioning AR to AV 0 Single Element Voltage Divider 0 Two Element Configuration 0 Signal Amplification 0 Input amp Output Loading Errors 4202009 Decibel Definition Amplifier Basics External Power 4202009 4202009 Example 1 Gain Amplifier An amplifier produces an output of 5 V when the input is 5 V What is the gain G and the decibel gain GdB Gain vuvin 5V5pV 1000000 GdB 20LogG 20Log1000000 120 dB Example 2 Gain Amplifier Ampli er has a gain of 60 dB lfthe input is 3 mV what is the output voltage GdB 20LogG or G Log1GdB20 Log16020 1000 VD Gvin 10003mV 3v 4202009 Input and Output Loading rrors Definition Input and Output Loading Errors Input and output loading are possible I roblems that can occur when using amplifiers and other signal conditioning devices Ifthe output voltage of the source device is altered when connected to a signal conditioning device then there exists a loading problem A simple model for Input and Output Loading Errors is RS R0 WrO WW 0 6 Vin Ri g VLoad g RLoad Q Q Transducer Amplifier Load Input and Output Loading Errors Typically you have a choice of using a noninverting or an inverting amplifier configuration Looking at the input an and output impedances of both configurations we can analyze the effects of the input and output loading errors The loading errors are defined as Input Loading Error RiRS Ri Output Loading Error RLoadRO RLoad RS R0 O lt Vin Ri w VLoad RLoad O O Transducer Amplifier Load 4202009 4202009 Example Input and Output Loading Errors NonInverting Amplifier Inverting Amplifier RS7OOQ RS7OOQ R 50m Ri12KQ R0 750 R0 750 Rm 1M0 Rm 1M0 R5 R0 437 V R w VLoad gt Mad Transducer Amplifier Load Nonlnverting Amplifier Error in 50K50K700 0986 This means that the amplifier is receiving 0986 x voltage coming from the transducer Error out 1M1M 75 0999 This means that the amplifier is delivering to the load 0999 x voltage it is amplifying Inverting Amplifier Error in 12K12K700 0945 This means that the amplifier is receiving 0945 x voltage coming from the transducer Error out 1M1M 75 0999 This means that the amplifier is delivering to the load 0999 x voltage it is amplifying Linearity Error Definition Linearity Error Lets say we characterized a load cell Assume the calibration plot is shown below Notice the slight honlinearity Load Cell Plot y 50106X 11697 R2 o 9903 Force lbs 4202009 4202009 Linearity Error The ideal linear plot is based on two points ofthe calibration plot the start point and the end point Linearity is the deviation between the measured data and the calculated ideal linear line Measured Data Load Cell Plot lmmmr 300 250 g 50 W 0 0 10 20 30 40 50 Force lbs Example Force Measuring system one and two element configurations Force Measuring System Application Two strain gages are mounted on a cantilever beam as shown below We wish to make a force measuring system capable of measuring a 010 lb force The resistance of each gage was measured over the entire input range and the results are illustrated in the data table Specify the necessary AR A V signal conditioning stage and amplifier type and gain necessary to calibrate the output voltage such that 1 lb 100mV F 010Ib Strain Gages Gage Raislznce iohms Force Measuring System Application Gage Resistance vs inmeoree Tensiun Cumressiun y2x35E E Input Farce llh 4202009 Force Measuring System Application SingleElement Configuration 5V SinglerEiemenl Voilage Dlvldel Circuit w Bipolar Suppiy Force Measuring System Application 1Gage System VB2EKRJGVR VQ VB 2E4ARRAR2 Bridge Sensitivity 1389mV10V Bridge Sensitivity 1389mVV V VOIVW 1000mV1389mV m 72 Required amplifier gain 4202009 Force Measuring System Application 1Gage System ompm Plot y 99 994x 4 2541 R1 n 9999 iGageSysiemOuipui Lmeav 1 Gage Sys12m0uipui ompm anlzge mVl E Input Farce in Force Measuring System Application 1Gage System Max Linearity Errur n 7n 4202009 Force Measuring System Application TwoElement Configuration RFEVB R2 is TworElement Voitage Divider CircuilwBipoiarSuppiy V V o BAJ Nonrlnvening mpii ier T 5v v Force Measuring System Application 2Gage System VB2EKRJGVR VQ VB 2E2ARR Bridge Sensitivity 2857mV10V Bridge Sensitivit 2857mVV AV VOIVm 1000mV2857mV Av 35 Required amplifier gain 4202009 Force Measuring System Application zGage syaem syaem OmputPlot v 1qu RI 1 ompm anllg mv Innn Fume um Force Measuring System Application 2Gage System 4202009 4202009 Input and Output Loading Errors NonInvertingAmglifier InvemngAmEIi er Rs7000 RS7000 R 50m R 12KQ RD 750 RE 750 RLuad 1MQ Rmad 1MQ RS RU Hui Way lt Vln Rlt w Load RLnad Transducer Amphfler Load Input and Output Loading Errors 2 G age System Lmeanty Errur Max Lmeanty Max Luamng Errur H mm Errur 4 39 4202009 Force Measuring System Application 2Gage System System ompumm vmm R39Zi a 2 Gage svsieinompm Wnn wading Lmeav 2 Gage svsmn output a lnrml Fame 1th Input and Output Loading Errors 1G age System Linearity Luading Errur Errur Max Linearity Max Luauing Ermr H mm Errur 71 43 TMAE 5132C Essentials oprplied Electrical Engineering Georgia Southern Universit Mechanical and Electrical Engineering Technology Spring 2009 Force Measuring System Prof AlbaFlores METEET Dept Generalized Measurement System 3 stages of a Measurement System Physical Parameter Sensor Signal Readout Transducer Conditioning Recording 4242009 4242009 Example of a Measuring System A Force Measuring System Using Strain Gages Displacement Measurement Systems F Cantiiever beam with a strain physicai gage mounted neartne base Parameter Wheatstone Strain Voitmeter Gage Sensor I 3 Readout Conditioning Strain 8 is the amount of deformation of a body due to an applied force e ALL Force Force 4 D L gt lt L gtlt ALH Strain can be positive tension or negative compression Units of strain inin or mmmm Because strain values are usually very small the microstrain us is mostly used 105 Strain Electrical Eguation 0 Strain of Strain Gage a 1GFARR 1 0 GFGage Factor 0 AR Change in Gage Resistance small quantities o R Nominal Gage Resistance Typical strain gages parameters are nominal resistance of 120 ohms and a gage factor of 21 o R 120 ohms 03FLl Force Measurements 4242009 Strai Mechanical Eguation Stress is the force per unit area A Temi mi 6 F A Where Ais the crosssect area G M y I see next slide F I I moment of inertia b h3 12 solid cuboid L MFLyh2 h bl G FLh2 bh312 6FL bhZ Young s Modulus E stressstrain E G 8 E Aluminum 10X106psi 2 mquot we F in 4i gtx L vlb When a force F is applied at the end ofthe beamthe stress 6 along the xaxis at is given by G M y I where M isthe bending moment M F L Nm y is the distance from the neutral axis of the beam Typically yh2 meters h beam thickness I isthe moment of inertia across section of the beam Ib h312 4242009 I Wheatstone Bridge consists of 4 resistors output voltage measured at the center when output voltage Vo 0 the bridge is balanced I One or more of the legs of the bridge may be a resistive transducer such as a strain gage I In the force measuring system the strain gage is in tension and it will be the R1 in the bridge Converting changes in resistance to changes in voltage E Wheats tone Bridge The output of the bridge V0 is given by R R v s 2 iRaR4 R R2i E 0 Notethat if R1R2R3R4 then Vo0 0 Under this condition the bridge is said to be balanced 0 Any change in resistance in any arm of the bridge will result in a nonzero output voltage 4242009 Force Measurement Experiment Setup 0 Consider R2 R3 and R4 fixed resistors with identical value R2 R3 R4 1209 3 0 R1 is the Strain Gage with a nominal resistance of 120 Q 0 When the gage is under a force then R1 i 120 Q Rl 120 AR and the bridge will be unbalanced and a voltage will appear across the middle of the bridge that can be measured with a DMlVI The bridge output voltage is predicted by the formula VO Rz R1 R2 R3 Rs R4 l E quotquot 4 where R1 120 AR 5 NOTE Be careful with the notation for Young s Modulus E eq 2 E Aluminum 10x106psi Power Supply for the Wheatstone Bridge E 5 volts eq 4 4242009 Force Measurements Homework 1 Using eqs 1 to 5 calculate the corresponding parameters given in the following table Indicate the units of F AR R1 RAR V0 1 2 3lb 5lb 7lb 10 b 2 Prove that for R1RAR R2R3 R4 R Vo7E4ARRAR2 The steps to calculate the parameters are Given F use eq 2 to compute 5 Using the value from step 1 use eq 1 and solve for AR Calculate the expected value for R1 for each force Using eqs 3 4 and S calculate Vo Prove that V0 E4AR R AR2 when R2R3R4R and R1 R AR 333355 4242009 Converting AR AV Wheatstone Bridge QuarterBridge Strain Gage Circuit or SingleElement Varying Bridge strain gage 4242009 4242009 QuarterBridge Strain Gage Circuit Now if we replace R4 for a Strain Gage any changes in the strain gage resistance will unbalance the bridge and produce a nonzero output voltage This is called quarter bridge configuration If the nominal resistance of the strain gage is RG and R R6 then the strain induced change in resistance AR is given by ARR ARR ALL 5 GFGage Factor 8 Strain inin AR RG GFE GF If R RG the bridge equation can be rewritten Note that the term 11GF 39 82 indicates the nonlinearity of the quarter bridge output V0 with respect to strain 8 4242009 Force Measurement Systems Cantilever beam with a strain gage mounted near the base Measurand Strain bridge or Amplifier voltmeter Gage voltage diVIder Stage curcunt Sensor l i S39gnal Readout Conditioning compression Half Bridge Configuration strain gage strain gage tension Iquot f Half Bridge Strain Gage Circuit HalfBridge Sensor Arrangement R 1 Rgagel H 1 Vu 39 Rgagez Bridge Gage In tension F R6 AR Gage in compression Rs AR tension RGAR R1R2RGR v01 GFE JEE V RG AR V0 2 R 4242009 strain gage Two Element Varying Bridge same leg V VEx AR Vo 39 E Vo 7 Output is linear strain gage strain gage Two Element Varying Bridge opposite leg 2 AR 0 39 2 AR R 7 Output is nonlinear 4242009 4242009 Full Bridge Fu brkigr saran n gauge circqu sxr in gauge 7 all air HausaLI sir asset strain 93 39u sham gauge 451mm 3 mast RGAR Four Element Varying Bridge opposite leg Vo GFe vEX V0 VEX Output is linear Full Bridge Cantilever Beam Setup 4242009 l R4 R2 R1 E R2 r 1 Va R1V1139R1R30R4V2 Vautl W Differential Low Pass 39 39 Filter C Transducer Setuo Sensitivity Definition for Transducers and Amplifiers Sensitivity Gage Factor Sensitivity sometimes referred to as gage factor is used to determine the output voltage of a transducer with a specified supply excitation voltage connected and a physical quantity applied Multiplying the excitation voltage by the sensitiviw yields the full scale output ofthe transducer Transducer 4r 10quot vquot Vcu39 quotV lexcwanon Sensmvily full scale max I amvrv 4242009 Sensitivity Definition for Transducers and Amplifiers conic 30m 0 1W y lexoiaipnl V V mil scale max in this gure with 10v excitation applied and a sensitivity of 3mvv a the tr nsducer has a maximum full scale output of anmv 10v 3mVV The instantaneous output voltage is directly proportional to the physical duantity applied For xample consider a memo psi pressure transducer that requires mv of excitation and has a sensitivityonmv Then if mu psi are applied the transducer will output znmv the full scale output if 50 psi are applied to thetransducer it will output 10mVhalf the full scale output 4242009 Measuremem Range ka2 swam gagebasee ransduzers amphhers a so have a sensmwwspeumamn deavaexmnsduzerandamphherwm sharexhesam nsmvwvaHnwmgmunhzememHmeasuremem uzer e 52 range at he ransd Dermmgme amph ersensmww sme mpuxvokage requved m mderm prnduze maxwmum eumm raxedvnkage mme amphher m m exammea 2 mV mm max m he amphherwmwe d 1v m he mm Dermmgme sensmwwrano R Transduzer SensvavAmphher Sensww ansduzerSensmVWAmphherSensmww 22 1 Dmmngmemu seaxe measuremem range nfmevansduzerbv R 10000 95 1 10000 95 Bezause s uzerand amph erhavexhesamesensmww human a he amphhens Lamb s eramphwmgmemu measuremennange orme ransduzer homo psw AzAz one in cases where the transducer has a higher sensitivitythan the amplilierthe transducer will be underutilized and the sensitivity ratio will be gt 1 iuuwp si Flessure Transdwer Sensitivity 3 mVN Calculating the sensitivity ratio R Transducer SensitivityAmplifier Sensitivity 3 2 15 Dividing the lull scale measurement range ol the transducer by R 0 psi 15 666667 psi Because the transducer has a higher sensitivity the amplilier cannot amplily the lull range 10000 psi ol thetransducer limiting the lull scale measurement range to 666667 psi I I I ma list the sensitivity ol a transducer as 2mVv in a catalog the actual sensitivity will be listed in the spec sheet as something like 209mvV or 214mVV rder to ensure the most accurate measurements these Slight ollsets must be taken into account during calibration To ensure that the transduceramplilier combination will accommodate the lull range ol data that y in acquire it may be necessary to oversize the transducer il you are measuring 0120 psi lor example choose a transducer with a 0 150 psi lull scale measurement range 4242009 Voltage Divider vs Bridge Configuration Can t use a single supply introduces a large offset approximately 12 the supply voltage The bipolar power supply is used The output is single ended meaning it is referenced to ground Bridge con guration output is a differential output Singleelement bridge configuration has the same sensitivity as the singleelement voltagedivider con guration Bridge Characteristics Brid eSensitivit mVV Bridge sensitivity is a ratio of the maximum expected change in output voltage to the bridge excitation voltage Wheatstone bridge only requires a single supply Has a differential output Twoelement bridge is twice as sensitive as a single element bridge Fourelement bridge is four times as sensitive as a singleelement bridge 4242009 19 TMAE 5132C Essentials of Applied Electrical Engineering Georgia Southern University Mechanical and Electrical Engineering Technology Spring 2009 Measurement Systems An example Prof AlbaFlores METEET Dept Generalized Measuremen System 3 stages of a Measurement System Physical Parameter Sensor Signal Readout Transducer Conditioning Recording Example of a Measuring System A Displacement Measuring System Using Strain Gages Displacement Measurement ystems Displacement Cantilever beam with a strain physical gage mounted near the base Parameter Wheatstone Strain Bridge or Amplifier Voltmeter voltage divider Gage Stage CIrCUIt i x J i Sensor S39gnal Readout Conditioning Background Strain Measurement Strain 8 is the amount of deformation of a body due to an applied force Force a ALL lt AL H Force Strain can be positive tension or negative compression Units of strain inin or mmmm Because strain values are usually very small the microstrain us is mostly used 105 The Sensor Metallic Strain Gage Very fine wire or metallic foil arranged in a grid pattern The grid pattern maximizes the amount of metallic foil subject to strain alignment marks I4 active grid gt length carrier Strain Gages The strain experienced by the test specimen is transferred directly to the strain gage which responds with a linear change in electrical resistance Nominal resistance values are in the range 30 to 30000 Being 120 Q 350 Q and 1000 Q the most common values Tension causes resistance increase Gage insensitive to Resistance measured lateral forces between these oints P Compression causes resistance decrease Parameters for choosing strain gages 39 Gage Length Accuracy requirements Maximum strain elongation Heat diSSipation 39 Gage Resistance Signal noise ratio Leadwire desenzitiazion Ga e Pattern g Heat dISSIpatIon Uniaxial biaxial or multiaxial Heat dissipation Strain gradients Installation space and easiness Gage Materials Gage wire sensitivity Adhesive Selftemperaturecompensation 0 It is very important that the strain gage be properly mounted onto the test specimen so that the strain is accurately transferred from the test specimen through the adhesive and strain gage backing to the foil itself 0 The Gage Factor GF is defined as the ration of fractional change in electrical resistance to the fractional change in length strain 2 ARR ARR GF AL L 8 0 A typical value for a gage factor is GF 20 The sina conditionin Converting AR gt AV In practice strain measurements involves very small values microstrain us 106 To have accurate measurements of these very small changes in resistance an adequate electrical circuit should be used The most common circuits are the voltage divider and the Wheatstone Bridge Voltaqe Divider Wheatstone Bridqe Converting AR gt AV gt Voltage Divider 0 Single power supply unipolar o Bipolar power supply gt Wheatstone Bridge 0 Quarter bridge configuration 0 Half bridge configuration 0 Full bridge configuration VoltageDivider Circuit Single Voltage Supply Voltage Divider Circuit wSingle Supply V0 ER2R1 R2 The strain gage changes its resistance with applied force can be connected as the R2 in the voltage divider circuit to convert changes in resistance to changes in voltage Ifwe select R1 RG where RG is the nominal resistance of the strain gage no force applied and R2 is the strain gage R2 RG AR then Vo E2 when no force is applied to the gage ie AR 0 Note that any variation in V0 from this reference value will indicate that a force is being applied to the gage VoltageDivider Circuit Bipolar Supply v0 2E SingleElement Voltage Divider w Bipolar Supply V0 R 21 R1R2 2 For the case 2E AR T RAR2 Note the nonlinear relation between V0 and AR Displacement Measurement System 51min gage l I Displacement 7 7 7 Cantilever beam with a strain Displacement mounted nearthe base Voltage Amplifier voltmeter Divider Stage Circuit Strain Gage Characterization Set the cantilever beam S rll I 39 ng g system and adjust It to be horizontal and tighten 1 Displacement Measure the strain gage resistance over the displacement range of interest Input Value Displacement Strain Gage Characterization cont The sign in the change of resistance AR will be positive if the strain gage resistance increases as you bend down the beam this means that the strain gage is in compression Otherwise if the change of resistance AR is negative it means that the beam is in tension this will occur ifthe beam is bent upwards strain was Z 1 Displacement Strain Gage Characterization c0nt Strain Gage Sensitivity From the data in the table a plot of resistance vs displacement for the gage can be obtained and the sensitivity Qmm of the strain gage can be found Change in Resistance vs Displacement Cycle 1 N Cycle2 Cycle 3 x Cycle4 L 0 Cycle 5 Average Linear Average Change in resistance 9 02 46810121416182022 Displacement mm From the plot Strain Gage Sensitivity Qmm 1 Qmm Note to obtain more accurate values the process of bending the beam and measuring its resistance for each position is repeated 5 times cycles Then the the average is calculated and a trendline through the data is obtained Then the sensitivity of the strain gage is calculated The siV nal conditionini Converting AR gt AV strain gage i I Displacement Displacement Cantilever beam with a strain mounted nearthe base Voltage Amplifier voltmeter Divider Stage Circuit Voltage Divider Stage 25V 25V R15 3209 VD 320 QAR I SingleElement Voltage Divider Circuit w Bipolar Supply SingleElement Voltage Divider Circuit w Bipolar Supply 2 1 V0 R1 R2 2 25V where R1 is fixed at 320 Q and R2 varies with displacement 25V Let R1 RC3 and R2 RG AR then V E L o 4 RAR2 Note Relation between V0 and AR is nonlinear Output is not linear However ifAR is small compared with R then the nonlinearity can be largely ignored Displacement Measuring System Measured Ca cu ated Calculate the expected output value using eqn 1 or eqn 2 R1RRGSZOQ R2RAR Eqn1 R 1 V2E 2 R1R2 2 Eqn2 2E AR V 4 R
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