Understanding Basic Analog Ideal Op Amps


 Barnaby Hart
 4 years ago
 Views:
Transcription
1 Appliction Report SLAA068A  April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp). It ssumes tht slient prmeters re perfect. Severl exmples of op mp circuits re described. Contents Introduction The Noninverting Op Amp The Inverting Op Amp The Adder The Differentil Amplifier Complex Feedbck Networks Video Amplifiers Cpcitors Conclusions List of Figures 1 The Noninverting Op Amp The Inverting Op Amp The Adder Circuit The Differentil Amplifier Differentil Amplifier With CommonMode Input Signl T Network in Feedbck Loop Thevenin s Theorem Applied to T Network Video Amplifier LowPss Filter HighPss Filter
2 Introduction The nme Idel Op Amp is pplied to this nd similr nlysis becuse the slient prmeters of the op mp re ssumed to be perfect. There is no such thing s n idel op mp, but present dy op mps come so close to idel tht Idel Op Amp nlysis becomes close to ctul nlysis. Op mps deprt from the idel in two wys. First, dc prmeters, such s input offset voltge, re lrge enough to cuse deprture from the idel. The idel ssumes tht input offset voltge is zero. Second, c prmeters, such s gin, re function of frequency, so they go from lrge vlues t dc to smll vlues t high frequencies. Both error sources re treted in lter Understnding. ppliction notes published in this series. This ssumption simplifies the nlysis, thus it clers the pth for insight. It is so much esier to see the forest when brush nd huge trees do not surround you. Although the idel op mp nlysis mkes use of perfect prmeters, the nlysis is often vlid becuse some op mps pproch perfection. In ddition, when working t low frequencies, severl khz, the idel op mp nlysis produces ccurte nswers. Voltge feedbck op mps re covered in this ppliction note, nd current feedbck op mps re covered in lter ppliction notes. Severl ssumptions hve to be mde before the idel op mp nlysis cn proceed. First, ssume tht the current flow into the input leds of the op mp is zero. This ssumption is lmost true in FET op mps where input currents cn be less thn pa, but this is not lwys true in bipolr highspeed op mps where tens of µa input currents re found. Second, the op mp gin is ssumed to be infinite, hence it drives the output voltge to ny vlue required to stisfy the input conditions. This ssumes tht the op mp output voltge cn chieve ny vlue. Sturtion occurs when the output voltge comes close to power supply ril, but relity does not negte the ssumption, it only bounds it. Also, implicit in the infinite gin ssumption is the need for zero input signl. The gin drives the output voltge until the voltge between the input leds (the error voltge) is zero. This leds to the third ssumption tht the voltge between the input leds is zero. The impliction of zero voltge between the input leds mens tht if one input is tied to hrd voltge source such s ground, then the other input is t the sme potentil. The current flow into the input leds is zero, so the input impednce of the op mp is infinite. Four, the output impednce of the idel op mp is zero. The idel op mp cn drive ny lod without n output impednce dropping voltge cross it. The output impednce of most op mps is frction of n ohm for low current flows, so this ssumption is vlid in most cses. Five, the frequency response of the idel op mp is flt; this mens tht the gin does not vry s frequency increses. By constrining the use of the op mp to the low frequencies, we mke the frequency response ssumption true. PARAMETER NAME PARAMETERS SYMBOL VALUE Input current IIN 0 Input offset voltge VOS 0 Input impednce ZIN Output impednce ZOUT 0 Gin 2 Understnding Bsic Anlog Idel Op Amps
3 The Noninverting Op Amp The noninverting op mp hs the input signl connected to its noninverting input, thus its input source sees n infinite impednce. There is no input offset voltge becuse V OS = V E = 0, hence the negtive input must be t the sme voltge s the positive input. The op mp output drives current into until the negtive input is t the voltge,. This ction cuses to pper cross. The voltge divider rule is used with being the input to the voltge divider, nd being the output of the voltge divider. Since no current cn flow into either op mp led, use of the voltge divider rule is llowed. Eqution 1 is written with the id of the voltge divider rule, nd lgebric mnipultion yields eqution 2 in the form for gin prmeter. VE Figure 1. The Noninverting Op Amp (1) 1 (2) When becomes very lrge with respect to, / 0 nd eqution 2 reduces to eqution 3. 1 Under these conditions = 1 nd the circuit becomes unity gin buffer. is usully deleted to chieve the sme results, nd when is deleted, cn lso be deleted ( must be shorted when it is deleted). When nd re deleted, the op mp output is connected to its inverting input with wire. Some op mps re selfdestructive when is left out of the circuit, so is used in mny buffer designs. When is included in buffer circuit, its function is to protect the inverting input from n over voltge, nd it cn hve lmost ny vlue (20k is often used). cn never be left out of the circuit in current feedbck mplifier design becuse determines stbility in current feedbck mplifiers. Notice tht the gin is only function of the feedbck nd gin resistors, so the feedbck hs ccomplished its function of mking the gin independent of the op mp prmeters. The gin is djusted by vrying the rtio of the resistors. The ctul resistor vlues re determined by the impednce levels tht the designer wnts to estblish. If = 10K nd = 10K the gin is two s shown in eqution 2, nd if = 100K nd = 100K the gin is still two. The impednce levels of 10 K or 100 K determine the current drin, the effect stry cpcitnce will hve, nd few other points. The impednce level does not set the gin; the rtio of / does. (3) Understnding Bsic Anlog Idel Op Amps 3
4 The Inverting Op Amp The noninverting input of the inverting op mp circuit is grounded. One ssumption we mde is tht the input error voltge is zero, so the feedbck keeps inverting the input of the op mp t virtul ground (not ctul ground but cting like ground). The current flow in the input leds is ssumed to be zero, hence the current flowing through equls the current flowing through. Using Kirchoff s lw, we write eqution 4. Algebric mnipultion gives us eqution 5. I1 I2 IB IB VE Figure 2. The Inverting Op Amp I 1 I 2 (4) (5) Notice tht the gin is only function of the feedbck nd gin resistors, so the feedbck hs ccomplished its function of mking the gin independent of the op mp prmeters. The ctul resistor vlues re determined by the impednce levels tht the designer wnts to estblish. If =10K nd =10K the gin is minus one s shown in eqution 5, nd if =100K nd =100K the gin is still minus one. The impednce levels of 10K or 100K determine the current drin, the effect stry cpcitnce will hve, nd few other points, but the impednce level does not set the gin; the rtio of / does. One finl note; the output signl is the input signl mplified nd inverted. The input impednce is set by becuse the inverting input led is held t virtul ground. The Adder An dder circuit cn be mde by connecting more inputs to the inverting op mp. The opposite end of the resistor connected to the inverting input is held t virtul ground by the feedbck; therefore, dding new inputs does not ffect the response of the existing inputs. V1 V2 R1 R2 VN RN Figure 3. The Adder Circuit Superposition is used to clculte the output voltges resulting from ech input, nd the output voltges re dded lgebriclly to obtin the totl output voltge. Eqution 6 is the output eqution when V 1 nd V 2 re grounded. Equtions 7 nd 8 re the other superposition equtions, nd the finl result is given in eqution 9. 4 Understnding Bsic Anlog Idel Op Amps
5 N R N V N (6) 1 V 1 2 V 2. V 1 V 2 R N V N. (7) (8) (9) The Differentil Amplifier The differentil mplifier circuit mplifies the difference between signls pplied to the inputs. Superposition is used to clculte the output voltge resulting from ech input voltge, nd then the two output voltges re dded to rrive t the finl output voltge. V1 R1 R2 V V V2 R3 R4 Figure 4. The Differentil Amplifier The op mp input voltge resulting from the input source, V 1, is clculted in equtions 10 nd 11. The voltge divider rule is used to clculte the voltge, V, nd the noninverting gin eqution (eqution 2) is used to clculte the noninverting output voltge, 1. V V 1 1 V (G ) V 1.. The inverting gin (eqution 5) is used to clculte the stge gin for 2 in eqution 12. These inverting nd noninverting gins re dded in eqution V 2.. (12) (10) (11) V 1.. V 2 (13) When = nd =, eqution 13 reduces to eqution 14.. V1 V 2. (14) Understnding Bsic Anlog Idel Op Amps 5
6 It is now obvious tht the differentil signl, (V 1 V 2 ), is multiplied by the stge gin, so the nme differentil mplifier suits the circuit. Becuse it only mplifies the differentil portion of the input signl, it rejects the commonmode portion of the input signl. A commonmode signl is illustrted in Figure 5. Becuse the differentil mplifier strips off or rejects the commonmode signl, this circuit configurtion is often employed to strip dc or injected commonmode noise off signl. V1 VCM V2 Figure 5. Differentil Amplifier With CommonMode Input Signl The disdvntge of this circuit is tht the two input impednces cnnot be mtched when it functions s differentil mplifier, thus there re two nd three op mp versions of this circuit specilly designed for high performnce pplictions requiring mtched input impednces. Complex Feedbck Networks When complex feedbck networks re put into the feedbck loop, the circuits get hrder to nlyze becuse the gin equtions cn not be used. The usul technique is to write node or loop equtions, nd to solve these equtions. Becuse component is grounded, superposition is not of ny use, but Thevenin s theorem usully cn be used s is shown in the exmple problem given below. Sometimes it is desirble to hve low resistnce pth to ground in the feedbck loop. Stndrd inverting op mps cn not do this when the driving circuit sets the input resistor vlue, nd the gin specifiction sets the feedbck resistor vlue. Inserting T network in the feedbck loop yields degree of freedom tht enbles both specifictions to be met with low dc resistnce pth in the feedbck loop. X R1 R2 Y R4 Figure 6. T Network in Feedbck Loop Brek the circuit t point X Y, stnd on the terminls looking into, nd clculte the Thevenin equivlent voltge s shown in eqution 15. The Thevenin equivlent impednce is clculted in eqution Understnding Bsic Anlog Idel Op Amps
7 V TH R TH (15) (16) Replce the output circuit with the Thevenin equivlent circuit s shown in Figure 7, nd clculte the gin with the id of the inverting gin eqution s shown in eqution 17. R1 R2 RTH VTH Figure 7. Thevenin s Theorem Applied to T Network Substituting the Thevenin equivlents into eqution 17 yields eqution 18. V TH R TH R TH... R3... (17) (18) Algebric mnipultion yields eqution 19. (19) Specifictions for the circuit you re required to build re n inverting mplifier with n input resistnce of 10K ( = 10K), gin of 100, nd feedbck resistnce of 20K or less. The inverting op mp circuit cn not meet these specifictions becuse must equl 1000K. Inserting T network with = = 10K nd = 485K does meet the specifictions. Video Amplifiers Video signls contin high frequencies, nd they use coxil cble to trnsmit nd receive signls. The cble connecting these circuits hs chrcteristic impednce of 75 Ω. To prevent reflections, which cuse distortion nd ghosting, the input nd output circuit impednces must mtch the 75 Ω cble. Mtching the input impednce is simple for noninverting mplifier becuse its input impednce is very high; just mke R IN = 75 Ω. nd cn be selected s very high vlues, in the kω rnge, so tht they hve miniml ffect on the impednce of the input or output circuit. A mtching resistor, R M, is plced in series with the op mp output to rise its output impednce to 75 Ω; terminting resistor, R T, is plced t the input of the next stge to mtch the cble. Understnding Bsic Anlog Idel Op Amps 7
8 RIN RM RT Figure 8. Video Amplifier The mtching nd terminting resistors re equl in vlue, nd they form voltge divider of 1/2 becuse R T is not loded. Very often is selected equl to so tht the op mp gin equls two. Then the system gin, which is the op mp gin multiplied by the divider gin, is equl to one (2 1/2 = 1). Cpcitors Cpcitors re key component in circuit designer s tool kit, thus short discussion on evluting their ffect on circuit performnce is in order. Cpcitors hve n impednce of X C = 1 (2πfC). Note tht when the frequency is zero the cpcitive impednce (lso known s rectnce) is infinite, nd tht when the frequency is infinite the cpcitive impednce is zero. These endpoints re derived from the finl vlue theorem, nd they re used to get rough ide of the ffect of cpcitor. When cpcitor is used with resistor, they form wht is clled brekpoint. Without going into complicted mth, just ccept tht the brek frequency occurs t f = 1/(2π RC) nd the gin is 3 db t the brek frequency. CF Figure 9. LowPss Filter The low pss filter circuit hs cpcitor in prllel with the feedbck resistor. The gin for the low pss filter is given in eqution 20. X C At very low frequencies X C, so domintes the prllel combintion in eqution 20, nd the cpcitor hs no effect. The gin t low frequencies is /. At very high frequencies X C 0, so the feedbck resistor is shorted out, thus reducing the circuit gin to zero. At the frequency where X C = the gin is reduced to hlf or 3 db becuse equl impednces in prllel equl hlf the vlue of either impednce. Connecting the cpcitor in prllel with where it hs the opposite effect mkes high pss filter. Eqution 21 gives the eqution for the high pss filter. (20) 8 Understnding Bsic Anlog Idel Op Amps
9 CG Figure 10. HighPss Filter 1 X C (21) At very low frequencies X C, so domintes the prllel combintion in eqution 21, nd the cpcitor hs no effect. The gin t low frequencies is 1 /. At very high frequencies X C 0, so the gin setting resistor is shorted out thus incresing the circuit gin to mximum. This simple technique is used to predict the form of circuit trnsfer function rpidly. Better nlysis techniques re presented in more dvnced ppliction notes for those pplictions requiring more precision. Conclusions When the proper ssumptions re mde, the nlysis of op mp circuits is strightforwrd. These ssumptions, which include zero input current, zero input offset voltge, nd infinite gin, re not n unrelistic ssumption becuse the new op mps mke them true in mny pplictions. When the signl is comprised of low frequencies, the gin ssumption is vlid becuse op mps hve very high gin t low frequencies. When CMOS op mps re used, the input current is in the femto mp rnge; close enough to zero for most pplictions. Lser trimmed input circuits reduce the input offset voltge to few micro volts; close enough to zero for most pplictions. The idel op mp is becoming rel; especilly for undemnding pplictions. The mth required for idel op mp nlysis is not rigorous, thus most people should be ble to nlyze simple op mp circuits. The more dvnced pplictions require complex op mp circuits, but there re mny of these shown in the pplictions literture. Grb TI op mp nd hve fun. Understnding Bsic Anlog Idel Op Amps 9
10 IMPORTANT NOTICE Texs Instruments nd its subsidiries (TI) reserve the right to mke chnges to their products or to discontinue ny product or service without notice, nd dvise customers to obtin the ltest version of relevnt informtion to verify, before plcing orders, tht informtion being relied on is current nd complete. All products re sold subject to the terms nd conditions of sle supplied t the time of order cknowledgment, including those pertining to wrrnty, ptent infringement, nd limittion of libility. TI wrrnts performnce of its semiconductor products to the specifictions pplicble t the time of sle in ccordnce with TI s stndrd wrrnty. Testing nd other qulity control techniques re utilized to the extent TI deems necessry to support this wrrnty. Specific testing of ll prmeters of ech device is not necessrily performed, except those mndted by government requirements. Customers re responsible for their pplictions using TI components. In order to minimize risks ssocited with the customer s pplictions, dequte design nd operting sfegurds must be provided by the customer to minimize inherent or procedurl hzrds. TI ssumes no libility for pplictions ssistnce or customer product design. TI does not wrrnt or represent tht ny license, either express or implied, is grnted under ny ptent right, copyright, msk work right, or other intellectul property right of TI covering or relting to ny combintion, mchine, or process in which such semiconductor products or services might be or re used. TI s publiction of informtion regrding ny third prty s products or services does not constitute TI s pprovl, wrrnty or endorsement thereof. Copyright 2000, Texs Instruments Incorported
2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationWeek 11  Inductance
Week  Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationSpace Vector Pulse Width Modulation Based Induction Motor with V/F Control
Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:
More informationEE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form example shown
EE247 Lecture 4 Ldder type filters For simplicity, will strt with ll pole ldder type filters Convert to integrtor bsed form exmple shown Then will ttend to high order ldder type filters incorporting zeros
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More information, and the number of electrons is 19. e e 1.60 10 C. The negatively charged electrons move in the direction opposite to the conventional current flow.
Prolem 1. f current of 80.0 ma exists in metl wire, how mny electrons flow pst given cross section of the wire in 10.0 min? Sketch the directions of the current nd the electrons motion. Solution: The chrge
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationTutorial on How to Create Electric Machine Models
PSIM Sotwre Tutoril on How to Crete Electric Mchine Models Powersi Inc. Septber 2009 www.powersitech.co Tutoril on Creting Electric Mchine Models Users cn crete electric chine odels using the bsic unction
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More informationHow To Set Up A Network For Your Business
Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More information4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.
4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.
More informationHow To Network A Smll Business
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationEngineertoEngineer Note
EngineertoEngineer Note EE265 Technicl notes on using Anlog Devices DSPs, processors nd development tools Contct our technicl support t dsp.support@nlog.com nd t dsptools.support@nlog.com Or visit our
More informationNetwork Configuration Independence Mechanism
3GPP TSG SA WG3 Security S3#19 S3010323 36 July, 2001 Newbury, UK Source: Title: Document for: AT&T Wireless Network Configurtion Independence Mechnism Approvl 1 Introduction During the lst S3 meeting
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationUplift Capacity of KSeries Open Web Steel Joist Seats. Florida, Gainesville, FL 32611; email: psgreen@ce.ufl.edu
Uplift Cpcity of KSeries Open Web Steel Joist Sets Perry S. Green, Ph.D, M.ASCE 1 nd Thoms Sputo, Ph.D., P.E., M.ASCE 2 1 Assistnt Professor, Deprtment of Civil nd Costl Engineering, University of Florid,
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationSolenoid Operated Proportional Directional Control Valve (with Pressure Compensation, Multiple Valve Series)
Solenoid Operted Proportionl Directionl Control Vlve (with Pressure Compenstion, Multiple Vlve Series) Hydrulic circuit (Exmple) v Fetures hese stcking type control vlves show pressure compensted type
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More information1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?
Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This printout should hve 22 questions, check tht it is complete. Multiplechoice questions my continue on the next column or pge: find ll choices efore mking your
More informationHow fast can we sort? Sorting. Decisiontree model. Decisiontree for insertion sort Sort a 1, a 2, a 3. CS 3343  Spring 2009
CS 4  Spring 2009 Sorting Crol Wenk Slides courtesy of Chrles Leiserson with smll chnges by Crol Wenk CS 4 Anlysis of Algorithms 1 How fst cn we sort? All the sorting lgorithms we hve seen so fr re comprison
More informationEconomics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999
Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,
More informationUnleashing the Power of Cloud
Unleshing the Power of Cloud A Joint White Pper by FusionLyer nd NetIQ Copyright 2015 FusionLyer, Inc. All rights reserved. No prt of this publiction my be reproduced, stored in retrievl system, or trnsmitted,
More informationDecision Rule Extraction from Trained Neural Networks Using Rough Sets
Decision Rule Extrction from Trined Neurl Networks Using Rough Sets Alin Lzr nd Ishwr K. Sethi Vision nd Neurl Networks Lbortory Deprtment of Computer Science Wyne Stte University Detroit, MI 48 ABSTRACT
More informationData replication in mobile computing
Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationIncreasing Q of Waveguide PulseCompression Cavities
Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide PulseCompression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque
More informationPHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS
PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationEasyMP Network Projection Operation Guide
EsyMP Network Projection Opertion Guide Contents 2 About EsyMP Network Projection Functions of EsyMP Network Projection... 5 Vrious Screen Trnsfer Functions... 5 Instlling the Softwre... 6 Softwre Requirements...6
More informationEngineertoEngineer Note
EngineertoEngineer Note EE280 Technicl notes on using Anlog Devices DSPs, processors nd development tools Visit our Web resources http://www.nlog.com/eenotes nd http://www.nlog.com/processors or emil
More informationClearPeaks Customer Care Guide. Business as Usual (BaU) Services Peace of mind for your BI Investment
ClerPeks Customer Cre Guide Business s Usul (BU) Services Pece of mind for your BI Investment ClerPeks Customer Cre Business s Usul Services Tble of Contents 1. Overview...3 Benefits of Choosing ClerPeks
More informationWEB DELAY ANALYSIS AND REDUCTION BY USING LOAD BALANCING OF A DNSBASED WEB SERVER CLUSTER
Interntionl Journl of Computers nd Applictions, Vol. 9, No., 007 WEB DELAY ANALYSIS AND REDUCTION BY USING LOAD BALANCING OF A DNSBASED WEB SERVER CLUSTER Y.W. Bi nd Y.C. Wu Abstrct Bsed on our survey
More informationRedistributing the Gains from Trade through Nonlinear. Lumpsum Transfers
Redistributing the Gins from Trde through Nonliner Lumpsum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lumpsum trnsfer rules to redistribute the gins from
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationTechniques for Requirements Gathering and Definition. Kristian Persson Principal Product Specialist
Techniques for Requirements Gthering nd Definition Kristin Persson Principl Product Specilist Requirements Lifecycle Mngement Elicit nd define business/user requirements Vlidte requirements Anlyze requirements
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationLecture 9 Microwave Network Analysis A. Nassiri  ANL June 19, 2003. Microwave Physics and Techniques UCSB June 2003 1
Lecture 9 Microwve Network nlysis. Nssiri  NL June 9, 003 Microwve Physics nd Techniques UC June 003 Prmeter Mesurement Technique VVM: The vector voltmeter mesures the mgnitude of reference nd test voltge
More informationModule Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationSmall Business Cloud Services
Smll Business Cloud Services Summry. We re thick in the midst of historic sechnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly
More informationPerformance analysis model for big data applications in cloud computing
Butist Villlpndo et l. Journl of Cloud Computing: Advnces, Systems nd Applictions 2014, 3:19 RESEARCH Performnce nlysis model for big dt pplictions in cloud computing Luis Edurdo Butist Villlpndo 1,2,
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationNQF Level: 2 US No: 7480
NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................
More informationInnovative and applied research on big data platforms of smart heritage
Innovtive nd pplied reserch on big dt pltforms of smrt heritge J. Qiu, J. Li, H. Sun * qiujie@thid.cn lijijun@thid.cn sunhuijio@thid.cn KEY WORDS: Smrt heritge, Big dt, Explntion ABSTRACT: Big dt hs huge
More informationFAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University
SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility
More informationAn Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process
An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct
More informationHillsborough Township Public Schools Mathematics Department Computer Programming 1
Essentil Unit 1 Introduction to Progrmming Pcing: 15 dys Common Unit Test Wht re the ethicl implictions for ming in tody s world? There re ethicl responsibilities to consider when writing computer s. Citizenship,
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationJaERM SoftwareasaSolution Package
JERM SoftwresSolution Pckge Enterprise Risk Mngement ( ERM ) Public listed compnies nd orgnistions providing finncil services re required by Monetry Authority of Singpore ( MAS ) nd/or Singpore Stock
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More information