Sirindhorn International Institute of Technology Thammasat University at Rangsit


 Leonard Ellis
 2 years ago
 Views:
Transcription
1 Sirindhorn Interntionl Institute of Technology Thmmst University t Rngsit School of Informtion, Computer nd Communiction Technology COURSE : ECS 204 Bsic Electricl Engineering L INSTRUCTOR : Asst. Prof. Dr. Prpun Suksompong WEB SITE : EXPERIMENT : 02 Network Theorems I: Thevenin & Norton Theorems. I. OBJECTIVES 1. To verify Thevenin s theorem for resistive circuits. 2. To verify Norton s theorem for resistive circuits. 3. To lern how to construct current source from the power supply. 4. To ecome fmilir with potentiometer. II. BASIC INFORMATION Let s consider liner circuit whose two terminls nd re connected to n ritrry lod. Thevenin s nd Norton s theorems ssert tht the circuit cn thus e replced y either Thevenin or Norton equivlent circuit, which cts like the originl circuit cross the lod connected to the two terminls. Thevenin nd Norton theorems re very useful in circuit nlysis for simplifying prts of complicted circuits. For resistive circuits, the Thevenin equivlent circuit (shown in Figure 21) simply consists of Thevenin voltge source V TH in series with Thevenin resistnce R TH, while the Norton equivlent circuit (shown in Figure 22) consists of Norton current source I N in prllel with Norton resistnce, which is the sme s R TH. V TH cn e determined from the opencircuit voltge cross terminls , i.e., the voltge cross the two terminls when the lod is disconnected. R TH is the equivlent resistnce of the circuit with respect to terminls  fter dectivting ll independent sources in the circuit nd disconnecting the lod. I N cn
2 e determined from the shortcircuit current t terminls , i.e., the current flowing through the shortcircuit connecting terminls . Reminders: (1) A voltge source is dectivted when it gives 0 V. In which cse, it ecomes short connection. (2) A current source is dectivted when it gives 0 A. In which cse, it ecomes n open connection. (3) We my use the terms turn off or disle insted of dectivte. They do not necessrily men powering off the power supply. II.1. Thevenin s theorem Thevenin s theorem provides method for simplifying circuit to stndrd equivlent form. As shown in Figure 21, the Thevenin equivlent circuit consists of ) Thevenin equivlent voltge source (VTH) in series with ) Thevenin equivlent resistnce (RTH). VTH nd RTH cn e found, when RLod is disconnected from nodes nd. The Thevenin voltge VTH is defined s the opencircuit voltge etween nodes nd. RTH is the totl resistnce ppering etween nd when ll sources re dectivted. This series comintion of VTH nd RTH is equivlent to the originl circuit in the sense tht if we connect the sme lod cross terminls  of ech circuit, we will get the sme voltge nd current t the terminls of the lod. This equivlence holds for ll possile vlues of lod resistnce. R TH V TH R Lod Figure 21: Thevenin equivlent circuit. 2
3 II.2. Norton s theorem The Norton equivlent circuit consists of n independent current source (IN) in prllel with n equivlent resistnce (RN), rrnged s shown in Figure 22. We cn simply derive this equivlent circuit from the Thevenin equivlent circuit y using the source trnsformtion concept. Thus, the Norton current ctully equls the shortcircuit current t the terminls , nd the Norton resistnce is identicl to the Thevenin resistnce. I N R N = R TH R Lod Figure 22: Norton equivlent circuit. Exmple: Figure 23 shows n exmple of circuit whose lod resistor is disconnected to nodes nd, creting n open circuit, so tht the Thevenin voltge VTH cn e mesured etween nodes nd (e.g. using voltmeter). In ddition, Figure 23 illustrtes the sme circuit, with the lod resistor disconnected, llowing mesurement of the Thevenin resistnce RTH (e.g. using n ohmmether cross nodes nd ) when ll voltge sources re replced with short circuit nd ll current sources re replced with n open circuit. Figure 23c shows n exmple of circuit whose lod resistor is disconnected from nodes nd nd then short circuit is creted etween the two nodes so tht Norton current IN cn e mesured (e.g. using n mpmeter). In ddition, the Norton equivlent resistnce RN cn e otined in similr mnner s RTH shown in Figure Figure 23d illustrtes the Thevenin nd Norton equivlent circuits derived in this exmple. 5 ohms 120 V 20 ohms 3 A 4 ohms + V TH =108V _ Figure 23: A circuit used for illustrting the Thevenin s nd Norton s theorems. 3
4 5 ohms 20 ohms 4 ohms R TH = R N = 8 ohms Figure 23: The circuit with the voltge nd the current sources dectivted to find RTH. 5 ohms 4 ohms 120 V 20 ohms 3 A I N = 13.5 A Figure 23c: The circuit with the shortcircuit etween  for finding the Norton current. 8 ohms R Lod 108 V 13.5 A 8 ohms R Lod Figure 23d: The Thevenin nd Norton equivlent circuits. Note tht VTH = IN RTH. III. MATERIALS REQUIRED  DC power supplies  Multimeters  Resistors (1/4W): 470, 1.2k, 10k, two of 330, nd potentiometer (vrile resistor). 4
5 IV. PROCEDURE Prt A: Thevenin equivlent circuit 1. Let R1= 330, R2 = 470, R3 = 330, nd RL = 1.2 k. Use DMM to mesure the resistnce of ech resistor, nd record the vlues in Tle Turn on the power supply, nd mesure its output voltge VPS. Adjust VPS to 12 V. Record the mesured vlue of VPS in Tle Connect the circuit in Figure 24. A R ohms B V ps 12V R ohms C R3 330 ohms R L 1200 ohms + V TH _ A D E Figure 24: The circuit for verifying Thevenin s nd Norton s theorems. The current IL through the lod is mesured y the mmeter A. 4. Mesure IL (the current through R L ), nd record this vlue in Tle 21, under the column Originl circuit. 5. Disconnect nd remove RL from the circuit, nd mesure the voltge cross node CE. This is V TH. Record the vlue in Tle 21 under the V TH Mesured column. 6. Turn off the power supply, nd disconnect it from the circuit. 7. Short AE y connecting wire cross node A nd E. 8. With RL still disconnected, mesure the resistnce cross nodes C nd E. This is R TH. Record the vlue in Tle 21 under the R TH mesured column. 5
6 9. Now we will uild the Thevenin equivlent circuit. The circuit shown in Figure 25 is our implementtion of Figure 21. To do this, djust the power supply so tht VPS = VTH, which hs een previously mesured. Connect DMM in the resistnce mesurement mode cross the potentiometer, nd djust its resistnce until the vlue of RTH is otined. Record the vlues of VTH (cross the power supple) nd RTH (cross the potentiometer) in Tle Connect the circuit s in Figure 25. This is the Thevenin equivlent circuit of the circuit in Figure 24. V PS =V TH R = R TH R L 1200 ohms A Figure 25: Thevenin equivlent circuit. The current IL through the lod is mesured y the mmeter A. 11. Mesure IL, nd record the vlue in Tle 21 under the Thevenin equivlent circuit, mesured column. Turn off the power supply. 12. Use the vlues of VPS, R1, R2, nd R3 to clculte VTH for the circuit in Figure 24. Record your result in Tle 21 under VTH clculted. 13. Clculte RTH in Figure 24 using the vlues of R1, R2, nd R3. Record your result in Tle 21 under RTH, clculted. 14. Use the clculted vlues of VTH nd RTH to clculte IL. Record the result under IL clculted. 6
7 Prt B: Norton equivlent circuit 1. From the vlue of IL nd RTH in Prt A, copy vlues of IL nd RTH in Tle 22, under "IL mesured, Originl circuit" nd "RN clculted," respectively. 2. Turn on the power supply, nd djust it ck to VPS = 12V. Record the mesured vlue of VPS in Tle 22. Connect the circuit in Figure 24 gin. To mesure IN, short circuit cross RL (from node C to node D), nd mesure the shortcircuit current. 1 Think out why RL does not hve to e removed. Record the vlue in Tle 22 under the column IN Mesured. 3. Now we will uild the Norton equivlent circuit. The circuit shown in Figure 26 is our implementtion of Figure 22. To do this, first djust the potentiometer until the resistnce vlue is equl to the vlue of RTH, mesured in Tle 21 recorded from step A.9. (It should lredy e t this vlue.) This will e your RN. Record the ctul (remesured) resistnce vlue cross the potentiometer in Tle 22 under the column RN Mesured. Now, for the current source, set the output of the power supply to its lowest vlue, i.e., 0 V. Connect the circuit shown in Figure 26. Meter A1 will e used to mesure the Norton current source IN in step 4. Meter A2 will e used in step B.5 to mesure the lod current IL. Note tht with one DMM, use it s Meter A1 here nd then use it s Meter A2 in step 5. A1 A2 I N V PS R = R N (R N = R TH ) R L 1200 ohms Figure 26: Norton equivlent circuit. Note tht in step B.2 the DMM is used s mmeter A1 to mesure the mount of current flowing out of the power supply. Then, it is used s mmeter A2 in step B.5. 1 Note tht this is the sme s putting n mmeter cross R L in Figure 24. The mmeter itself cts s the short circuit cross R L nd it lso displys the vlue of the shortcircuit current. Alterntively, to mesure I N, we cn simply replce R L with n mmeter. 7
8 4. Turn on the power supply nd slowly increse the output of the supply until the current mesured y DMM A1 is equl to the vlue of IN, which hs een previously mesured. Record the vlue of IN tht you get from DMM A1 in Tle 22. Also, Record the power supply voltge VPS tht chieves this vlue of IN in Tle Record the lod current IL mesured y DMM A2 in Tle 22 under the Norton equivlent circuit, mesured column. Turn off the power. 6. Clculte the vlue of IN from the circuit of Figure 24, nd record your result in Tle 22 under IN clculted. 7. Clculte y using the clculted vlues of I N nd R N. Record your result in Tle 22 under, clculted. Remrk: Since the current source is not ville in the l, this prt of the experiment is modified to suit our ojective y djusting VPS in order to otin the current IN. Tips: From now on, when you re sked to use current source, replce it y power supply (voltge source) nd connect the rest of the circuit. Then, djust the vlue of the output voltge of the power supply so tht the required mount of current psses through it. Cution: Becuse power supply is not true current source, when you mke ny chnge to the circuit connection, the vlue of the current tht pss through the voltge source my chnge. You will need to redjust the voltge vlue of the power supply (voltge source) so tht the required mount of current psses through it every time tht you mke ny chnge to the circuit. Cution 2: DO NOT connect the DMM in its mmeter mode directly to the power supply. The mount of current coming out of the power supply when there is nothing connected to it except the DMM is meningless. 8
9 Tle 21: Thevenin equivlent circuit Mesured: R1 = R2 = R3 = RL = VPS = (A.2) V TH R TH Mesured Clculted Mesured Clculted Originl circuit Mesured Thevenin equivlent circuit A.5 A.12 A.8 A.13 A.4 A.11 A.14 Clculted A.9 A.9 TA Signture: Tle 22: Norton equivlent circuit VPS = (B.2) VPS = (B.4) I N R N Mesured Clculted Mesured Clculted Originl Circuit Mesured Norton equivlent Circuit Clculted B.2 B.6 B.3 B.1 B.1 B.5 B.7 B.4 9 TA Signture:
10 V. QUESTIONS 1. Consider the circuit given in Figure 24. If R 3 is replced y resistor, find the following vlues. V TH = V R TH = = ma If R L is lso replced y 120 resistor, determine the voltge nd current t R L. 2. Wht re the dvntges of using Thevenin s theorem nd Norton s theorem in solving complicted liner circuits? Show some exmples to support your nswers. 3. In step 2 of prt B, we tried to mesure IN. Why cn we leve RL in the circuit? 10
Version 001 CIRCUITS holland (1290) 1
Version CRCUTS hollnd (9) This printout should hve questions Multiplechoice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire
More information2 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 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 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 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 informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
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 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 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 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 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 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 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 information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
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 informationVariable Dry Run (for Python)
Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 2050 minutes
More informationUnderstanding Basic Analog Ideal Op Amps
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).
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 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 informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
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 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 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 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 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 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 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 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 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 informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
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 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 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 informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
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 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 information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
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 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 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 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 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 informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
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 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 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 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 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 informationQuick Reference Guide: Onetime Account Update
Quick Reference Guide: Onetime Account Updte How to complete The Quick Reference Guide shows wht existing SingPss users need to do when logging in to the enhnced SingPss service for the first time. 1)
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 informationRotational Equilibrium: A Question of Balance
Prt of the IEEE Techer InService Progrm  Lesson Focus Demonstrte the concept of rottionl equilirium. Lesson Synopsis The Rottionl Equilirium ctivity encourges students to explore the sic concepts of
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 informationWelch Allyn CardioPerfect Workstation Installation Guide
Welch Allyn CrdioPerfect Worksttion Instlltion Guide INSTALLING CARDIOPERFECT WORKSTATION SOFTWARE & ACCESSORIES ON A SINGLE PC For softwre version 1.6.5 or lter For network instlltion, plese refer to
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 informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
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 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 informationIFTA 23rd Annual Conference. Friday, October 8, 2010. Yukitoshi Higashino, MFTA
IFTA 23rd Annul Conference Fridy, Octoer 8, 2010 Border Line Pttern for the Ichimoku Kinko Hyo (Cndlestick Chrt)  Positioning with Bse Lines nd Timing for TimeBsed Loss Cutting 1 Yukitoshi Higshino,
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
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 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 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 informationSolution to Problem Set 1
CSE 5: Introduction to the Theory o Computtion, Winter A. Hevi nd J. Mo Solution to Prolem Set Jnury, Solution to Prolem Set.4 ). L = {w w egin with nd end with }. q q q q, d). L = {w w h length t let
More informationMATLAB Workshop 13  Linear Systems of Equations
MATLAB: Workshop  Liner Systems of Equtions pge MATLAB Workshop  Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB
More informationFirm Objectives. The Theory of the Firm II. Cost Minimization Mathematical Approach. First order conditions. Cost Minimization Graphical Approach
Pro. Jy Bhttchry Spring 200 The Theory o the Firm II st lecture we covered: production unctions Tody: Cost minimiztion Firm s supply under cost minimiztion Short vs. long run cost curves Firm Ojectives
More informationSection 5.2, Commands for Configuring ISDN Protocols. Section 5.3, Configuring ISDN Signaling. Section 5.4, Configuring ISDN LAPD and Call Control
Chpter 5 Configurtion of ISDN Protocols This chpter provides instructions for configuring the ISDN protocols in the SP201 for signling conversion. Use the sections tht reflect the softwre you re configuring.
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 informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationChapter 5: Elasticity. measures how strongly people respond to changes in prices and changes in income.
Chpter 5: Elsticity Elsticity responsiveness mesures how strongly people respond to chnges in prices nd chnges in income. Exmples of questions tht elsticity helps nswer Wht hppens to ttendnce t your museum
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 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 informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
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 informationThe Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,
More information4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A
Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter
More informationThe remaining two sides of the right triangle are called the legs of the right triangle.
10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right
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 informationVectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
More informationHomework #6 Chapter 7 Homework Acids and Bases
Homework #6 Chpter 7 Homework Acids nd Bses 18. ) H O(l) H 3O (q) OH (q) H 3 O OH Or H O(l) H (q) OH (q) H OH ) HCN(q) H O(l) H 3O (q) CN (q) H 3 O HCN CN Or HCN(q) H (q) CN (q) H CN HCN c) NH 3(q) H O(l)
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 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 informationMatrix Inverse and Condition
Mtrix Inverse nd Condition Berlin Chen Deprtment of Computer Science & Informtion Engineering Ntionl Tiwn Norml University Reference: 1. Applied Numericl Methods with MATLAB for Engineers, Chpter 11 &
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 informationFormal Languages and Automata Exam
Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider
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 informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More information6.5  Areas of Surfaces of Revolution and the Theorems of Pappus
Lecture_06_05.n 1 6.5  Ares of Surfces of Revolution n the Theorems of Pppus Introuction Suppose we rotte some curve out line to otin surfce, we cn use efinite integrl to clculte the re of the surfce.
More informationHomework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.
Text questions, Chpter 5, problems 15: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed
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 informationRepeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re shorthd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
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 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 informationCOMPLEX FRACTIONS. section. Simplifying Complex Fractions
58 (66) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours. 00 600 6 FIGURE FOR EXERCISE 9 95. Selling. George sells one gzine suscription every 0 inutes, wheres Theres
More informationModule 5. Threephase AC Circuits. Version 2 EE IIT, Kharagpur
Module 5 Threehse A iruits Version EE IIT, Khrgur esson 8 Threehse Blned Suly Version EE IIT, Khrgur In the module, ontining six lessons (7), the study of iruits, onsisting of the liner elements resistne,
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 information