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Measurement & Instrumentation

by: Ms. Isobel Rau

Measurement & Instrumentation EE 521

Ms. Isobel Rau

GPA 3.67


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This 47 page Class Notes was uploaded by Ms. Isobel Rau on Thursday October 15, 2015. The Class Notes belongs to EE 521 at New Mexico Institute of Mining and Technology taught by Staff in Fall. Since its upload, it has received 59 views. For similar materials see /class/223649/ee-521-new-mexico-institute-of-mining-and-technology in Electrical Engineering at New Mexico Institute of Mining and Technology.

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Date Created: 10/15/15
Using tricks of physics to measure The following concerns a question in a physics degree exam at the University of Copen hagen Describe how to determine the height of a skyscraper with a barometer One student replied You tie a long piece of string to the neck of the barometer then lower the barometer from the roof of the skyscraper to the ground The length of the string plus the length of the barometer will equal the height of the building This highly original answer so incensed the examiner that the student was failed The student appealed on the grounds that his answer was indisputably correct and the university appointed an independent arbiter to decide the case The arbiter judged that the answer was indeed correct but did not display any noticeable knowledge of physics To resolve the problem it was decided to call the student in and allow him six minutes in which to provide a verbal answer which showed at least a minimal familiarity with the basic principles of physics For ve minutes the student sat in silence forehead creased in thought The arbiter reminded him that time was running out to which the student replied that he had several extremely relevant answers but couldn7t make up his mind which to use On being advised to hurry up the student replied as follows Firstly you could take the barometer up to the roof of the skyscraper drop it over the edge and measure the time it takes to reach the ground The height of the building can then be worked out from the formula H gtz But bad luck on the barometer Or if the sun is shining you could measure the height of the barometer then set it on end and measure the length of its shadow Then you measure the length of the skyscraper7s shadow and thereafter it is a simple matter of proportional arithmetic to work out the height of the skyscraper But if you wanted to be highly scienti c about it you could tie a short piece of string to the barometer and swing it like a pendulum rst at ground level and then on the roof of the skyscraper The height is worked out by the difference in the gravitational restoring Or if the skyscraper has an outside emergency staircase it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths then add them up lf you merely wanted to be boring and orthodox about it of course you could use the barometer to measure the air pressure on the roof of the skyscraper and on the ground and convert the difference in millibars into feet to give the height of the building But since we are constantly being exhorted to exercise independence of mind and apply scienti c methods undoubtedly the best way would be to knock on the janitor7s door and say to him 7lf you would like a nice new barometer I will give you this one if you tell me the height of this skyscraper7 The student was Niels Bohr the only person from Denmark to win the Nobel prize for Physics force T 27139 Laplace transform Wikipedia the free encyclopedia lofl4 httpenwikipediaorgwikiLaplaceitransform Laplace transform From Wikipedia the free encyclopedia In mathematics the Laplace transform is a powerful technique for analyzing linear time invariant systems such as electrical circuits harmonic oscillators optical devices and mechanical systems to name just a few Given a simple mathematical or functional description of an input or output to a system the Laplace transform provides an alternative functional description that often simpli es the process of analyzing the behavior of the system or in synthesizing a new system based on a set of speci cations The Laplace transform is an important concept from the branch of mathematics called functional analysis In actual physical systems the Laplace transform is often interpreted as a transformation from the time domain point of view in which inputs and outputs are understood as functions of time to the frequency domain point of view where the same inputs and outputs are seen as functions of complex angular frequency or radians per unit time This transformation not only provides a fundamentally different way to understand the behavior of the system but it also drastically reduces the complexity of the mathematical calculations required to analyze the system The Laplace transform has many important applications in physics optics electrical engineering control engineering signal processing and probability theory The Laplace transform is named in honor of mathematician and astronomer Pierre Simon Laplace who used the transform in his work on probability theory The transform was discovered originally by Leonhard Euler the proli c eighteenth century Swiss mathematician Contents I 1 Formal de nition I 11 Bilateral Laplace transform l 12 Inverse Laplace transform l 2 Region of convergence l 3 Properties and theorems l 31 Laplace transform of a function39s derivative l 32 Relationship to other transforms l 321 Fourier transform l 322 Mellin transform l 323 Z transform l 324 Borel transform l 325 Fundamental relationships I 4 Table of selected Laplace transforms l 5 s Domain equivalent circuits and impedances l 6 Examples How to apply the properties and theorems l 61 Example 1 Solving a differential equation I 62 Example 2 Deriving the complex impedance for a capacitor l 63 Example 3 Finding the transfer function from the impulse response I 64 Example 4 Method of partial fraction expansion l 65 Example 5 Mixing sines cosines and exponentials 08222006 1215 PM Laplace 2of14 tiansform 7 Wikipedia the free encyclopedia hnpenWikipediaorgWildLaplaceiu ansform l 66 Example 6 Phase delay I 7 References l 8 See also I 9 External links Formal de nition The Laplace transform of a function f0 defined for all real numbers t2 0 is the function F defined by my c ma iffycam ntir him 1 112 function 6t lim 75 of th at 39T39h ln rlimit Mo 15 EI 0 if there is such an impulse inft at 0 The parameter s is in general complex 5 J lb This integral transform has a number of properties that make it useful for analysing linear dynamical systems The most v v v v H v v andimmmm 39 39 39 and ivi inn re nmi e wi his is similar to the way that logarithms change an o eration of multiplication of numbers to addition of their logarithms This changes integral equations and differential equations to polynomial equations which are much easier to solve Bilateral Laplace transform a or onersided transform is normally intended When one says quotthe Laplace transformquot without qualification the unilater Th 39 39 39 117 r 7 transfmm or tworsided Laplace transform by r extending the limits of39 39 a i If L 39 4 becomes a special case of th d f39 39 39 of the function by the Heaviside step function The bilateral Laplace transform is defined as follows 00 Fm 1 mm 752le dt Inverse Laplace transform TM in r I 39 whichis oi enb39 ft quotlFsll 1 i 00 st 2 m e Fslds 1 where 7 is a real number so that the contour path of integration is in the region of convergence of F s normally requiring q gt 1355 for every singularity Sp of Fs and z 71 If all singularities are in the left halfrplane that is ResPl lt D for every 5p then W can be A 39 39 formula 39 39 39 inverse Fourier transform for the inverseT by formula 082220061215 PM Laplace transform 7 Wikipedia the free encyclopedia impHenwudpediaorgwudLapiacejansrom Region of convergence The Laplace transform F typically exists for all complex numbers such that Re gt a where a is a real constant which depends on the growth behavior offt whereas the two7sided transform is defined in a range a lt Re lt 17 The subset of values ofsfor M h th T r 39 39 39 r rm nf or the domain of convergence In the two7sided case it is sometimes called the strip ofconvergence There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken other than to say the defining integral converges It is however easy to give theorems on cases where it may or may not be taken Properties and theorems Given the functions f0 and got and their respective Laplace transforms F and 65 fa 7 Ems gm 7 flaw the following is a list of properties of unilateral Laplace transform l Linearity C aftt bgt 7 ms has I Frequency differentiation tft 7 7F39sl Ctquotft 7 71quotFquots l Differentiation Am 7 mm 7 for CW 7 fan 7sf0 l 7mm c W 7 Mm 7 WWW 7 7 f H tU l I Frequency integration 7 Faudg I Integration A Mm dr 7 A mm M 7 ins I Scaling 2 man 7 EF 3 l Initial value theorem f0 SlggsFts 3 of14 082220061215 PM Laplace transform 7 Wildpedia the free encyclopedia impenwllclpedlaorgwlldLaplacejansfom l Final value theorem fltxgt Lin 5175 all poles in leftrhand plane The final value theorem is useful because it gives the longrterm behaviour without having to perform partial fraction d compositions or other difficult algebra If a functions poles are in the right hand plane eg eI or sint the 39 d behaviour of this formula is unde me I Frequency shifting e ft Fseal cquot Fs 7 ad z ft l Time shifting 5 ft 7 aut 7 a 5135 57197 Fsgt ft 7 ma 7 a Note 140 is the Heaviside step function I Convolution MN 90 Fs 65 l Periodic Function period T r f177nA e ftdt Laplace transform of a function s derivative It is often 39 quot 39 39 property of the T r 39 39 of afunction39s derivative This can be derived from the basic expression for a Laplace Transform as follows c mm Aime Si dt meet 3 oc 00 gist eDi isf mdt byparts u 7 f 0 i v e 7 gamt yielding a sum 7 f0 And in the bilateral case we have c sme39stftdt 5 1mm Relationship to other transforms 4 of 14 082220061215 PM Laplace transform 7 Wikipedia the free encyclopedia 50f14 impHenwudpediaorgwudLapiacejansrom Fourier transform The continuous ruuiiei 39 39 39 39 T with complex argument 5 to FM Hm 57M 175 5 mm m 2quot quotftdt Note that this expression excludes the scaling factor 1 f which is often included in definitions of the Fourier transform r T anla and wule 39 4 4 39 39 j spectrum ofa signal or dynamical system Mellin transform Th 39 d d T by a simple change of Variables If in the Mellin transform 657Mg9n7mesg9 we set 9 exp 7 t we get a tworsided Laplace transform Ztransf39orm The Zrtransform is simply the Laplace transform of an ideally sampled signal with the substitution of where T 39lfS is the sampling period in units of time eg seconds and f is the sampling rate in samples per second or hertz m Arm 2 i m 7 nT M be a sampling impulse train also called a Dirac comb and mm 7 11 i 6t 7717 710 7 i mm 771139 7 i 1mm 7 17 M Wu be the continuousrtime representation of the sampled rm In 2 I011quot are the discrete samples of It The Laplace transform of the sampled signal 110 is 082220061215 PM Laplace tiansform 7 Wikipedia the free encyclopedia 60f14 impHenwudpediaorgwudLapiacejansrom qus qte 5idt with the substitution of AL 57quot nm39nm 1 1 v itIr the sampled signal XqsXlelT Bore transform Theintegralform ofth 39 39 39 T r 39 J The enerali ed quot L T exponential type Fundamental relationships Since an ordinary Laplace transform can be written as a special case of a tworsided transform and since the tworsided transform can be written as the sum of two onersided transforms the theory of the Laplace Fourierr Mellinr and transforms at 39 quotbi t However quot 39 39 and quot 39 39 associated wi each of these four major integral transforms Table of selected Laplace transforms The following table provides Laplace transforms for many common functions of a single variable For definitions and explanations see the Exp anaton Note at the end of the tab e Because the Laplace transform is a linear operator 39T39h T r 1 f0 90 L39 mm 5 91 of 39 39 T eachterm The Laplace transform of a multiple of a function is that multiple times the Laplace tranformation of that function 5 afttll at lfltll Laplace transforms are only valid when t is greater than 0 7 which is why everything in the table below is a multiple of ut Here is a list of common transforms 082220061215 PM Laplace transform 7 Wikipedia the free encyclopedia 7of14 impHenwugipediaorgwudLapiacejansrom Region of Time Domain Frequency Domain ID Function conver ence rm 5 Was Xsgt L mm g for causal systems 1 ideal delay 6t 7 T 2 1a unit impulse 6 t 1 all 5 delayed nth power with t 7 739quot 7W7 e 2 7 t 7 gt 0 frequency shift n E u T 5 0M1 5 tquot 1 0 2a nth power a W SM 5 gt t 1 t 1 2a1 qthpower m 3914 Su s gt D 1 2a2 unit step ut 5 gt 3 2b delayed unit step t i T E 5 gt 0 2c ramp t39ut s gt D tquot M 2d nth power with frequency shift e 39 um 5 gt Q 2dl exponential decay 2quot ut s gt in 3 exponential approach 1 i 97 ut 5 gt D 4 sine sinwt ut 5 gt D 5 cosine 005th 39 H S gt D 6 hyperbolic sine sinhat ut s gt M 7 hyperbolic cosine Ema 39 0 s gt a 8 Exponentiallyrdecaying Sim I u S gt in e wave Exponentiallyrdecaying m S a t t 7 s gt 7a 9 cosine wave 5 COSW u 5 y quot 2 l 10 nth root WWW OHMquot391 1 Z s gt D t in 11 natural logarithm 111E ut mam 7 5 gt D Besselfunction a 7 in 739 V D 12 of the firstldnd Mm um M quot549 of ordern S 2 08222006 1215 PM 80f14 Laplace uansform 7 Wikipedia the free encyclopedia impenWikipediaorgWikiLaplaceiu ansform s Domain equivalent circuits and impedances Th I r 39 J 39 rirrm t anal 139 an imnl conversions to the eromain of circuit elements can be made Circuit elements can be transfo med into impedances very similar to phasor impedances However eromain impedances are valid for many more inputs than phasor impedan s Here is a summary of equivalents Time Domain sDomain V R sameasnuvmal a I V 1 C V 1 Va V SC DHe sc 5 E I a El a I I v 1 V SL L 1 5L LI 1 gt m IE I I In 3 r i i v v v i v n M n v yummy vvvi conditions on the circuit elements For example if a capacitor has an initial voltage across it or if the inductor has an initial 39 39 39 39 39 quot 39 quotquot l 39 for nu mmd 39 derived Examples How to apply the properties and theorems e Laplace transform is used frequently in engineering and physics the output of a linear dynamic system calculated by convolving its unit impulse respons with the in t 39 orming this calculation in Laplace space turns the convo ution into a multiplication the latter being easier to solve because of its algebraic form For more information see control theory Th 1 r1 1 4 diff 139 ntial 39 4 39 extensively in electrical engineering The following examples derived from application in physics and engineering will use SI units afmeasnre s1 is based on metersfor distance in a m a em as m ii e and a pm a Example 1 Solving a differential equation The following example is based on concepts from nuclear physics Consider the following firstrorder linear differential equation 082220061215 PM Laplace iansform 7 Wikipedia the free encyclopedia impenWikipediaorgWildLaplaceiu ansfoim 7 4N where r 39 oflmdera ed a amnle of at tim in seconds and A is the decay constant We can use the Laplace transform to solve this equation Rearranging the equation to one side we have Next we take the Laplace transform of both sides of the equation 5N9 DAN5 0 where Ms HMO Solving we find Ms Finally we take the inverse Laplace transform to find the general solution Mm Nazi which is indeed the correct form for radioactive decay Example 2 Deriving the complex impedance for a capacitor 77m W W quotquot 39 39 39 nverning th 39 39 U r 39 following differential equation d 1 Ci ere C is the capacitance in farads of the capacitor 1 i0 is the electrical current in amperes owing through the capacitor as a function of time and v vt is the voltage in volts across the terminals of the capacitor also as a function of time Taking the Laplace transform of this equation we obtain 15 Csws 7 v 9 of14 082220061215 PM Laplace transform 7 Wikipedia the free encyclopedia hnpenWikipediaorgWildLaplaceiu ansfoim where 15 5M0 Vs cum and Solving for V we have I v 5 7 W a i tion of the complex impedance Z in ohms is the ratio of the complex Voltage Vdivided by the complex current I while holding the initial state V at zero 0 1 5639 which is the correct expression for the complex impedance of a capacitor Example 3 Finding the transfer function from the impulse response 77m example is based an concept from signal processing and describes the dynamic behavior afa damped hanmmic oscillator See also RLC circuit Time domam xm Consider a linear timerinvariant system with impulse response l l T quotverse I lama Ada are 1 Mt Ae cosmdt 7 31 t 7quot such that Mal quumcy unmam Wit 7 Chi Z 0 Rela onship between the me where t is the time in seconds and domain and me frequency domin 0 S w S 2 is the phase delay in radians Suppose that we want to find the transfer function of the system We begin by noting that ht Ae eos ant 7 to ut 7 td Md is the time delay of the system in seconds and ut is the Heaviside step function 10 of14 08222006 1215 PM Laplace uansform 7 Wikipedia the free encyclopedia impHenwudpediaorgwudLapiacejansrom 7 7 3 5a Hm 7 wimp42 Mums in 7 Ag 52 2515 a2 m5 5a A in a E 53 2 4mg where is m mun m 39 439 per second Example 4 Method of partial fraction expansion Consider a linear timerinvariant system with transfer function 1 H The impulse response is simply the inverse Laplace transform of this transfer function W CquotHs T 39 39 cuegln by nartial Hm i t i 5asm 5a 53 for unknown constants P and R To find these constants we evaluate 7 1 VHS and Substituting these values into the expression for 115 we find 1 1 1 H 7 7 7 7 S 3 in 5 m 5 3 Finally using the linearity property and the known transform for exponential decay see Item 3 in the Table of Laplace Transfonns above we can take the inverse Laplace transform of 115 to obtain hm Ems a i a W3 2 110f14 082220061215 PM Laplace uansform 7 Wikipedia the free encyclopedia impHenwudpediaorgwudLapiacejansrom which is the impulse response of the system Example 5 Mixing sines cosines and exponentials Time function Laplace transform L3 7 a 5 3 M e Cosmf m 5mmt 0ng Starting with the Laplace transform s L MS 7 s 12 e we find the inverse transform by first adding and subtracting the same constant a to the numerator S 0 sa2w3 7 70 MS sa2w2 By the shiftrinrfrequency property we have 1tequot i quot 5 Jr 524mg 59 7 7M r 5 3 0 w 2 n mm m MM 7 rai 7 5 393 71 W 4 in Wm M gta will Finally using the Laplace transforms for sine and cosine see the table above we have In of 0mm 3 0 smu1t Example 6 Phase delay Time function Laplace transform s sinagt woos o t 5m a w 52 w WWW soossfiwasmq m Starting with the Laplace transform X 5 51nd mfosa 5 we J k 12 of14 08222006 1215 PM Laplace uansform 7 Wikipedia the free encyclopedia impHenwudpediaorgwudLapiacejansrom ssinq mcos 52 sgm2 V 5 0 WC c2 smm We are now able to take the inverse Laplace transform of our terms 7 V 1 s 1 w 11 ch 52w2coso Sum smmcom suwtcos Xs To simplify this answer we must recall the trigonometric identity that asiwt bcoswt xa2 I2 mm arcta1ba and apply it to our Value for xt 1m loos2 a sin2 0 sm wt m ctan 5 39 C05 0 5111th w We can apply similar logic to find that Smjijw M References l A D Polyanin and A V ManzhiroV Handbook ofIntegral Equations CRC Press Boca Raton 1998 ISBN 0784937287674 I William McC Siebert Circuits Signals and Systems MIT Press Cambridge Massachusetts 198 6 ISBN 0726271922972 See also I Differential equations l Linear timerinvariant systems I Fourier transform l Integral transform l Pierre Simon Laplace l Leonhard Euler l Exponential function I Complex number External links I Online Computation I J r M inverse transform wimsunicefr 13 of14 082220061215 PM Laplace transform Wikipedia the free encyclopedia httpenwikipediaorgwikiLaplaceitransform l Tables of Integral Transforms httpeqworldipmnetruenauxiliaryauX inttranshtm at Equrld The World of Mathematical Equations l Eric W Weisstein Laplace Transform httpmathworldwolframcomLaplaceTransformhtml at MathWorld l Laplace Transform Table httpwwwefundacommathlaplacetransformtablecfm at eFunda Engineering Fundamentals l Laplace and Heaviside httpwwwintmathcomLaplacel alapunitstepfnsphp at Interactive maths Retrieved from quothttpenwikipediaorgwildlaplacetransform Categories Integral transforms Differential equations I This page was last modified 0717 21 August 2006 I All text is available under the terms of the GNU Free Documentation License See Copyrights for details Wikipedia is a registeied trademark of the Wildmedia Foundation Inc 14 of 14 08222006 1215 PM Prndu Bulletin 017999 June 1996 PIN Silicon Photodiode Type OP999 1 uxuam us e we mum Femunzs Nanwvecewmuanme s nsevs mamance F351 wncmnuume 3 c package sMe p an The oPaaa phutumude cunsw s ma PW smcun phutumude mggmeg m a davk pme wash warmquot Named sheH package Thenanuwvecemnuanme PvuvmeszxceHemunastcuummu T e sensuvs ave1 pmggcugn t2512d 1m Muse cune almn mp Omek GaA As Emmevs Omek s packagmg pvucess pmvmes ekceuem upucax ang mecpamcax 3st The sheH a su pmvmes i ekceuem upucax wens su acE cumvm g1 cmp mammary ang cunsw encv uHhe umsme package mmensmns Ahsglme Maximum Ratings 725 c un ess mkewse nmeg Revevse Eveakduwn vguage an v smvage ang Opevawvg Tempevamve Range ran 0 m 1 nn 0 Leag Su dekuTempevamve11Bmchl1 6 mm wgmcase 1m 5 sec mp su devmu nun zen c Puwev Dwssmauun mu mm E mkkwmmenaea Dgamnanbeex enaeammsc Menkwsgema Max mamas heappheamcaamnsaema 2 Devate hneanw 57 MW 2 apgve 25 aLapWaanmesacapksemaaaaeapeacaspeak Weak gag nm ang swam 25 mvwcmi weamuswn szeamm sumacmmnaeeaaemacaw mam Typical Perlnrmznce Cums Relmwe Respnnse vs Wmelenmh mm ml mum outrun Culll lVWaVE WW quotm Dwstance BEN227v LensTmsr mches OmekTechnmugV nc mavv CVUSW Ruad an 3232mm Fax an 3232395 Type OP999 Electrical Characteristics TA 25 C unless otherwise noted SYMBOL PARAMETER MIN TYP MAX UNITS TEST CONDITIONS IL Reverse Light Current 65 15 A vR 5 v E6 025 mWcm23 ID Reverse Dark Current 1 60 nA VR 30 V Ee 0 VBR Reverse Breakdown Voltage 60 V IR 100 pA VF Forward Voltage 12 V IF 1 mA CT Total Capacitance 4 pF VR 20 V Ee 0 f 10 MHZ tr tf Rise Time Fall Time 5 ns VR 20 V 7 850 nm RL 50 Q Typical Performance Curves Normalized Light Current vs Reverse Voltage rA 0 A935 nm ormalized to p 5 V Normalized Light Current 5 VR Reverse Voltage V Light Current vs lrradiance VR5V TA25 C m A89 nm 9 Ee Irradiance mWcm2 CT Total Capacitance pF NOTE SEE ABOVE FUR Total Capacitance vs Reverse Voltage VR Reverse Voltage V SNitching Time Test Circuit II39 K CONDITIONS VR Nomalizad Dark Current Nomalized Current Normalized Light and Dark Current vs Ambient Temperature 239n TA Ambient Temperature 0C Light Current vs Angular Displacement 935nm 5V e Angular Displacement Deg Optek reserves the right to make changes at any time in order to improve design and to supply the best product possib e Carrollton Texas 75006 Optek Technology Inc 1215 W Crosby Road 367 972323 2200 Fax 9723232396 FHOTOSENSORS Vp 5 V E s 7 9 m Normalized to m 395 TA 250 C O 1353 sI Light Cur ent 39 390 139 Q g m 95 g z 3 Dark Current I I I I I 55 a a n m n n m Lecture 5 September 13 2006 Analysis of Random Uncertainties and Resistive Sensors Analysis of Random Uncertainties Recap from last time Random versus systematic uncertainty Mean value Standard deviation Standard deviation of mean Resistive Sensors 39 How to measure resistance Wheatstones bridge Sensors Temperature sensors Strain Gauges Photoconductors Relative humidity sensors Linearangular position sensing Giant Magnetoresistive Effect Anisotropic Magnetoresistive Effect Resistive Temperature Sensors Resistance varies with temperature Some materials more than others Measure resistance compute temperature R 39 We can model RT as a polynomial RT ai T Toy T RT RTo a T To 6 T To2 Temperature coefficient Resistive Temperature Sensors Metals versus semiconductors Small versus large temperature coefficient Large temperature coefficient gives more precise temperature measurement Because we are measuring change in resistance Table 61 Resistive Temperature Sensors Diodes as resistive temperature sensors Diode current and voltage varies with temperature HPWPE 39WPE silicon silicon Catode Anode Cathode 0N voltage Hf quot 65if for Si If Anode Forward hiae Breakdown region A IPIU I v Reverse current 4121 for Ge 3 LA Note The reverse currehi R uerse shown J39s iypieeror39 type MOW For other Gives refer r to the respective detesheet Image not to scare Resistive Temperature Sensors 39 Applications Temperature measurements Measure airfluid flow Wheatstone39s Bridge Used to measure resistance V0 VB V0 C Vo RXR2 R1R3 VA VD R3 RXR1 R2 Balance bridge change RX and V0 is proportional to change in RX The uncertainty in ARX is then proportional to the uncertainty in V0 Resisitive Temperature Sensors 39 Eliminating lead resistances Several Wheatstone bridge designs Bridge Output 39Ftresistansne R1 Element Bridge Dulput Resistive Strain Gauges Effect Measure deformation of or forces on an object Mechanism Strain gauges change resistance when stretched Bonded Attached to object and deform with object Unbonded Used to measure forces acting on an object Strain is deformation caused by stress Resistive Strain Gauges 39 Stress and Strain 7 Stress is the force on the object stretching force 7 Strain is the resulting deformation of the object 7 Hooke39s law spring equation valid for small stress in elastic regionquot F k Resistive Strain Gauges 39 Stress Strain curves 7 One stress for each strain multiple strains for each stress Stress 2 Aluminum Resistive Strain Gauges Computing ARR as a function of force or strain Gauge factor Resistance varies linearly with strain constant of proportionality is material dependent Resistance change is small How do we measure it L L P AR AA A AL A2 A M A pL Q a g g e R A R A 0 L Photoconductors Mechanism Resistance decreases when material is illuminated 39 Different from photodiodessolar cells PhotodiodesSolar cells are pn junctions which produce a V or current when illuminated Photoconductors are passive resistive elements whose resistance depends on illumination Photoconductors 39 Resistance 39 Assignment 1 i 1 1 Figure 66 derive R T E R P expression for V0 in terms of Pi Photoconductors 39 Reminder Description of opamps and mathematical model Relative humidity sensing Effect Conductivity between two contacts changes when humidity of surrounding air changes Dunmore Mechanism Salt coated rod wound with metal wires Salt absorbs water releases ions which increase conductivity Chemical process Relative humidity sensing 39 Dunmore problems Migration of ions at high voltages which eventually increases resistance Charging like a battery Chemical breakdown Dunmore Solution Low voltage and AC signal Wheatstone bridge will still work because frequency is small and there are no large capacitances Relative humidity sensing 39 Practical use Limited useful range of RH sensing Widerange instrument may require combining multiple sensors with different useful ranges Relative humidity sensing Brady Mechanism Water absorption in crystal lattice Changes conduction band structure Increased RH decreases resistance Physical process New technology May replace chemistrybased RH sensors Linearangular position sensing Mechanism Position sensed by voltage division Linear voltage divider for linear position measurements Helical voltage divider for angular position sensing 39Issues Resistance must vary linearly with position or at least in a known way with position Temperature coefficient must be small or known and temperature must be measured Mechanical friction eventually wears resistor down Linearangular position sensing 39 Alternatives Linear Variable Differential Transformers Various optical encoding schemes Why not use these alternatives all the time Giant Magnetoresistive Effect Effect Resistance of a conducting layer sandwiched between two magnetic layers changes depending on the relative orientations of the Bfield in the two magnetic layer Giant Magnetoresistive Effect 39 Why the name Ordinary Magnetoresistance In some materials the resistivity can WM be lowered by few percent by applying a magnetic field rem cnw l Giant Magnetoresistance fx H In some layers of films mm X M resistance can be lowered ij f by tens of percent by a r applying a magnetic field 7 if 7 7 x 80 o m o H KGauss Giant Magnetoresistive Effect Mechanism Film of conducting material sandwiched between films of magnetic materials Measure resistance of conducting material Chargecarrier motion most restricted when fields are antiparallel High resistance Chargecarrier motion less restricted when fields parallel low resistance Lowest energy state is antiparallel fields Imposing external field parallellizes film fields and lowers resistance Giant Magnetoresistive Effect 39Issues Resistance varies nonlinearly with applied field Hysteresis Both complicate field strength measurements 39 Applications Discrete field strength measurements New miniaturization for magnetic storage devices Anisotropic Magnetoresistive Effect Effect Resistance varies with angle between conductor and applied magnetic field Linear when arranged as a Wheatstones bridge Can be used as electronic compass PIN Photodiodes Panasonic PNZ331 F PIN Photodiode For optical fiber communication systems I Features 0 Metal package with sllield pin a High coupling capability suitable for plastic ber and glass ber 0 High quantum ef ciency o Highspeed response 3420 450 04 I Absolute Maximum Parameter Reverse Power I Dimensions of detection area Unit mm 1 1 0 88 Activereglon A1 lt 001 A J I EIectroOptical Characteristics Ta 25 C Parameter Symbol Conditions min typ max Unit Dark current 1D VR 10V 01 10 nA Photo current IL VR 10V L 1000 1Xquot1 4 7 HA Peak sensitivity wavelength AP VR 10V 900 nm Frequency characteristics fcquotZ VR 10V RL 509 50 MHZ Capacitance between pins Q VR 10V 3 pF Photodetection sensitivity R VR 10V 9 800nm 045 055 AW Acceptance half angle 0 Measured from the optical falls to the half powerpolrlt 40 deg Photodetection surface shape D Effective photodetection area D 088 mm Note 1 Spectral sensitivity Sensitivity at wavelengths exceeding 400 nm as a percentage is 100 to maximum sensitivity Note 2 This product is not designed to withstand electromagnetic radiation or heavycharge particles Note 3 The glass strength of this product cannot withstand loads of05 kg or greater This fact needs to be taken into consideration if optical bers are to be mounted on the product 1 Measurements were made using a tungsten lamp color temperature T 2856K as a light source 2 Switching time measurement circuit see figure below Note Detection photo current 73 dB Sig 1N VR 10v 41 4 rnput pulse ta Delay time Mr 800nm 4 t Rise time Time required for the collector photo current 5393 OUT Output um to increase from 10 to 90 ofits nal value 509 RL p 1 tr Fall time Time required for the collector photo current to decrease from 90 to 10 ofits initial value Panasonic 1 PN2331 F Dark current rD nA Power dissipation PD mW Relative photo current rL rm PDiTa 20 40 60 80 1 00 Ambient temperature Ta c 1D VR 1 1 0 4 1 0 4 0 8 16 24 32 Reverse voltage VR V ILiTa N o oo o a o 0 40 20 0 80 100 Ambient temperature Ta quot3 Photo current IL uA Illuminance L 1x ILiTa Relative photo current 1L quot4 0 40 20 0 20 40 60 80 100 Ambienttemperature Ta Cc Spectral sensitivity characteristics 1 00 Relative sensitivity 5 quotn 0 200 400 600 800 1000 Wavelength 1 nm 1200 PIN Photodiodes Photo current IL uA IDiTa Dark current ID nA o 104 103 740 720 0 20 40 60 80 loo Ambient temperature Ta Cc Frequency characteristics 55 Ii 55 i w o IIIIllll IIIIIIIIIIIIIIIIIIIIII39ll IIII I IIIIIIIIIIIIII 4 tquot 1 10 10 2 103 Frequency f MHZ Relative power output P dB l l oo o rlo Panasonic PIN Photodiodes PN2331 F Coupling loss characteristics Coupling loss characteristics LxLY7XY LxLY7XY 0 E A 71 A 71 0 3 5 3 a s 72 s S f Ax 0 in in E E o E U 74 U U 5 i 5 08 04 0 04 08 08 04 0 04 08 Reverse voltage VR V Distance XY mm Distance X Y mm Coupling loss characteristics Coupling loss characteristics 0 l N l Coupling loss LZ dB Coupling loss LZ dB Distance 2 mm Distance 2 mm Panasonic 3


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"Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Anthony Lee UC Santa Barbara

"I bought an awesome study guide, which helped me get an A in my Math 34B class this quarter!"

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"


"Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

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